objective Ques (163 results)
SSC CGL Mains 20241)In a circle with centre O, an arc ABC subtends an angle of 134° at the centre of the circle. The chord AB is produced to a point P. ∠CBP is equal to:
67°
SSC CGL Mains 20242)
14 cm
SSC CGL 20223)In the figure, XYZ is a secant and ZT is a tangent to the circle at T. If TZ = 12 cm and YZ = 8 cm, then find the length of XY.
10 cm
SSC CGL 20224)The diameters of two circles are 12 cm and 20 cm, respectively and the distance between their centres is 16 cm. Find the number of common tangents to the circles.
3
SSC CGL 20225)If two circles of radii 18 cm and 8 cm touch externally, then the length of a direct common tangent is:
24 cm
SSC CGL 20226)If C1, C2 be the centres of two circles and r1, r2 be the respective radii such that the distance between the centres is equal to the sum of the radii of the two circles, find the number of common tangents.
3
SSC CGL 20227)Radius of a circle is 5 cm. Length of chord AB in this circle is 6 cm. What is the distance of this chord from the centre of the circle?
4 cm
SSC CGL 20228)The hour hand moves through 4 hours and has a length of 6 cm. Find the area (in cm2, rounded off to two decimal places) of the sector covered by the hour hand.
37.71
SSC CGL 20229)The circumference of the two circles is 110 cm and 330 cm respectively. What is the difference between their radii?
35 cm
SSC CGL 202210)Select the INCORRECT statement with respect to the properties of a circle.
The perpendicular distance from the centre of a circle increases when the length of a chord increases.
SSC CGL 202211)The length of the chord of a circle is 24 cm, and the perpendicular distance between the centre and the chord is 5 cm. The radius of the circle is:
13 cm
SSC CGL 202212)A chord of length 42 cm is drawn in a circle of diameter 58 cm. Another chord of length 40 cm is drawn parallel to the chord of length 42 cm. Find the difference between the distances of the two chords from the centre.
1 cm
SSC CGL 202213)Two equal circles of radius 8 cm intersect each other in such a way that each passes through the centre of the other. The length of the common chord is:
\(8\sqrt3 cm \)
SSC CGL 202214)O is the centre of this circle. Tangent drawn from a point P, touches the circle at Q. If PQ = 24 cm and OQ = 10 cm, then what is the value of OP?
13 cm
SSC CGL 202215)Radius of a circle is 10 cm. Angle made by chord AB at the centre of this circle is 60 degree. What is the length of this chord?
10 cm
SSC CGL 202216)The circumference of the two circles is 198 cm and 352 cm respectively. What is the difference between their radii?
24.5 cm
SSC CGL 202217)Two circles touch each other externally at P. AB is a direct common tangent to the two circles, A and B are points of contact, and ∠PAB = 40° . The measure of ∠ABP is:
50°
SSC CGL 202218)AB and CD are two chords in a circle with centre O and AD is the diameter. When produced, AB and CD meet at the point P. If ∠DAP = 27°, ∠APD = 35°, then what is the measure (in degrees) of ∠DBC?
28
SSC CGL 202219)AB is the diameter of a circle with centre O. C and D are two points on the circle on either side of AB, such that ∠CAB = 52° and ∠ABD = 47°. What is the difference (in degrees) between the measures of ∠CAD and ∠CBD?
10
SSC CGL 202220)In a circle with centre O, PQ, and QR are two chords such that ∠PQR = 118°. What is the measure of ∠OPR?
28°
SSC CGL 202221)AB is a chord of a circle with centre O. C is point on the circle in the minor sector. If ∠ABO = 50°, then what is the degree measure of ∠ACB ?
140°
SSC CGL 202222)O is the centre of a circle of radius 10 cm. P is a point outside the circle and PQ is a tangent to the circle. What is the length (in cm) of PQ if the length OP is 26 cm?
24
SSC CGL 202223)The radii of two concentric circles with centre O are 26 cm and 16 cm. Chord AB of the larger circle is tangent to the smaller circle at C and AD is a diameter. What is the length of CD?
38 cm
SSC CGL 202224)AC is the diameter of a circle dividing the circle into two semicircles. ED is a chord in one semicircle, such that ED is parallel to AC. B is a point on the circumference of the circle in the other semicircle. ∠CBE = 75°. What is the measure (in degrees) of ∠CED?
15°
SSC CGL 202225)AB is a chord in the minor segment of a circle with center O. C is a point between A and B on the minor arc AB. The tangents to the circle at A and B meet at the point D. If ∠ACB = 116°, then the measure of ∠ADB is
52°
SSC CGL 202226)Points A and B are on a circle with centre O. PA and PB are tangents to the circle from an external point P. If PA and PB are inclined to each other at 42°, then find the measure of ∠OAB.
21°
SSC CGL 202227)In a circle with centre O, PA and PB are tangents to the circle at point A and point B, respectively. C is a point on the major arc AB. If ∠ACB = 50°, then find the measure of ∠APB.
80°
SSC CGL 202228)In a circle with centre O and of radius 13 cm, two parallel chords are drawn on different sides of the centre. If the length of one chord is 10 cm and the distance between the two chords is 17 cm, then find the difference in lengths of the two chords (in cm).
14
SSC CGL 202229)AB is a chord of a circle with centre O. C is a point on the circumference of the circle in the minor sector. If ∠ABO = 40°, what is the measure (in degree) of ∠ACB?
130°
SSC CGL 202230)Chords AB and CD of a circle intersect externally at P. If AB = 7 cm, CD = 1 cm and PD = 5 cm, then 50% of the length of PA (in cm) is:
5
SSC CGL 202231)In the following figure, MN is a tangent to a circle with centre O at point A. If BC is a diameter and ∠ABC = 42°, then find the measure of ∠MAB.
48
SSC CGL 202232)PQ and RS are two parallel chords of a circle of length 14 cm and 48 cm, respectively, and lie on the same side of the centre O. If the distance between the chords is 17 cm, what is the radius (in cm) of the circle?
25
SSC CGL 202233)AB is a diameter of a circle with centre O. The tangent at a point C on the circle and AB, when produced, meet at the point P. If ∠APC = 38∘, then what is the measure of ∠PCB?
26°
SSC CGL 202234)A circle with centre O has radius 15 cm. D is a point on the circle such that a 24 cm long chord AB is bisected by OD at point C. Find the length of CD (in cm).
6
SSC CGL 202235)AB is a chord of a circle with centre O, while PAQ is the tangent at A.R is a point on the minor arc AB. If ∠BAQ = 70° then find the measure of ∠ARB .
110°
SSC CGL 202236)In the following figure, P and Q are centers of two circles. The circles are intersecting at points A and B. PA produced on both the sides meets the circles at C and D. If ∠CPB = 100°, then find the value of x.
100
SSC CGL 202237)An isosceles ΔMNP is inscribed in a circle. If MN = MP = 16√5 cm, and NP = 32 cm, what is the radius (in cm) of the circle?
20
SSC CGL 202238)In the given figure, O is the centre of the circle. ∠POQ = 54°. What is the measure (in degree) of ∠PRQ?
153
SSC CGL 202239)Chords AB and CD of a circle intersect externally at P. If AB = 7 cm, CD = 1 cm and PD = 5 cm, then the length of PB (in cm) is:
3
SSC CGL 202240)Chords AB and CD of a circle, when produced, meet at the point P. If AB = 6.3 cm, BP = 4.5 cm, and CD = 3.6 cm, then the length (in cm) of PD is
5.4 cm
SSC CGL 202241)In a circle of diameter 20 cm, chords AB and CD are parallel to each other. BC is diameter. If AB is 6 cm from the centre of the circle, what is the length (in cm) of the chord CD?
16
SSC CGL 202242)In a circle with centre O, AC and BD are two chords. AC and BD meet at E, when produced. If AB is a diameter and ∠AEB = 36°, then the measure of ∠DOC is:
108°
SSC CGL 202243)In a circle, ABCD is a cyclic quadrilateral. AC and BD intersect each other at P. If AB = AC and ∠BAC = 48°, then the measure of ∠ADC is
114°
SSC CGL 202244)Two common tangents AC and BD touch two equal circles each of radius 7 cm, at points A, C, B and D, respectively, as shown in the figure. If the length of BD is 48 cm, what is the length of AC?
50 cm
SSC CGL 202245)A tangent is drawn from a point P to a circle, which meets the circle at T such that PT = 10.5 cm. A secant PAB intersects the circle in points A and B. If PA = 7 cm, what is the length (in cm) of the chord AB?
8.75
SSC CGL 202246)AB is the diameter of a circle with centre O. C and D are two points on the circumference of the circle on either side of AB, such that ∠CAB = 42° and ∠ABD = 57°. What is difference (in degrees) between the measures of ∠CAD and ∠CBD?
18°
SSC CGL 202247)Two circles touch each other externally at T. RS is a direct common tangent to the two circles touching the circles at P and Q. ∠TPQ = 42°. ∠PQT (in degrees) is:
48
SSC CGL 202248)In a circle with centre O, chords PR and QS meet at the point T, when produced, and PQ is a diameter. If ∠ROS = 42º, then the measure of ∠PTQ is
69º
SSC CGL 202249)O is the centre of a circle with diameter 20 cm. T is a point outside the circle and TA is a tangent to a circle. If OT is 26 cm, what is the length (in cm) of the tangent TA?
24
SSC CGL 202250)AB is a diameter of a circle with centre O. A tangent is drawn at point A. C is a point on the circle such that BC produced meets the tangent at P. If ∠APC = 62º, then find the measure of the minor arc AC.
28º
SSC CPO 202051)The circles of radii 15 cm and 10 cm intersect each other and the length of their common chord is 16 cm. What is the distance (in cm) between their centres?
\(6 + \sqrt {161}\)
SSC CPO 202052)In a circle with centre O, AD is a diameter and AC is a chord. Point B is on AC such that OB = 7 cm and ∠OBA = 60°, If ∠DOC = 60°, then what is the length of BC?
7 cm
SSC CPO 202053)PA and PB are two tangents from a point P outside the circle with centre O. If A and B are points on the circle such that ∠APB = 142°, then ∠OAB is equal to:
71°
SSC CPO 202054)Chord AB of a circle is produced to a point P, and C is a point on the circle such that PC is a tangent to the circle. If PC = 12 cm, and BP = 10 cm, then the length of AB (in cm) is:
4.4
SSC CPO 202055)PA and PB are two tangents from a point P outside the circle with center O at the point A and B on it. If ∠APB = 130°, then ∠OAB is equal to:
65°
SSC CPO 202056)PA and PB are two tangents from a point P outside the circle with centre O. If A and B are points on the circle such that∠APB = 100°, then∠OAB is equal to:
50°
SSC CPO 202057)In a ΔABC, the bisectors of ∠B and ∠C meet at O. If ∠BOC = 142°, then the measure of ∠A is:
104°
SSC CPO 202058)PA and PB are two tangents from a point P outside the circle with centre O. If A and B are points on the circle such that ∠APB = 128°, then ∠OAB is equal to:
64°
SSC CPO 202059)A circle is inscribed in a triangle ABC. It touches sides AB, BC and AC at points R, P and Q, respectively. If AQ = 3.5 cm, PC = 4.5 cm and BR = 7 cm, then the perimeter (in cm) of the triangle ΔABC is:
30
SSC CHSL 202160)Two equal circles of radius 8 cm intersect each other such that each passes through the centre of the other. The length of the common chord is:
8√3 cm
SSC CHSL 202161)Two circles with centres O and P and radii 17 cm and 10 cm, respectively, intersect each other at A and B. The length of the common chord AB is 16 cm. What is the perimeter of the triangle OAP (in cm)?
48
SSC CHSL 202162)ΔABC is drawn in a circle such that AC = BC and ∠BAC = 65°. From points B and C two tangents are drawn which intersect at point P. What is the measure of ∠BPC?
50°
SSC CHSL 202163)O is the centre of a circle of radius 9 cm. M is a point outside the circle and MN is a tangent to the circle. What is the length (in cm) of OM if the length MN is 12 cm?
15
SSC CHSL 202164)In a circle with centre O and radius 6.5 cm, a chord AB is at a distance 2.5 cm from the centre. If tangents at A and B intersect at P, then find the distance of P from the centre.
16.9 cm
SSC CHSL 202165)Chords AB and CD of a circle meet at point P (outside the circle), when produced. If AB = 9 cm, PB = \(\frac{1}{3}\)AB and CD = 5 cm, then the length of PD (in cm ) is:
4
SSC CHSL 202166)Two parallel chords are drawn in a circle of diameter 50 cm on the opposite sides of its centre. The length of one chord is 40 cm and the distance between the two chords is 22 cm. The length of the other chord is:
48 cm
SSC CHSL 202167)Line AC is a tangent to a circle at point B on it, and PQ is a chord of the circle such that BP = BQ. If ∠ABP = 64°, then find the measure of ∠PBQ.
52°
SSC CHSL 202168)A, B and C are three points on a circle whose centre is O. If angle BOC is equal to 124°, then what is the value (in degrees) of angle BAC ?
62
SSC CHSL 202169)From a point P, which is at a distance of 13 cm from the centre O of a circle, a pair of tangents PQ and PR of length 12 cm are drawn to the circle. The area of the quadrilateral PQOR (in cm2) is:
60
SSC CHSL 202170)A 9-cm-long perpendicular is drawn from the centre of circle to a chord of length 24 cm. The radius of the circle is:
15 cm
SSC CHSL 202171)\(\frac{775 \ \times \ 775 \ \times \ 775 \ + \ 225 \ \times \ 225 \ \times \ 225}{77.5 \ \times \ 77.5 \ + \ 22.5 \ \times \ 22.5 \ - \ 77.5 \times \ 22.5}\) is equal to:
100000
SSC CHSL 202172)Two circles of radii 18 cm and 12 cm interest each other and the length of their common chord is 16 cm. What is the distance (in cm) between their centres?
2√5 (2 + √13)
SSC CHSL 202173)If the length of a chord of a circle, that makes an angle of 60° with the tangent drawn at one end point of the chord, is 8√3 cm, then the radius of the circle will be:
8 cm
SSC CHSL 202174)In a circle with center O, PA and PB are tangents to the circle at A and B, respectively, from an external point P. If, ∠AOB = 116° then what is the measure of ∠OPB?
32°
SSC CHSL 202175)Let O be the centre of a circle and AC be the diameter. BD is a chord intersecting AC at E. AD and AB are joined. If ∠BOC = 40° and ∠AOD = 120°, then ∠BEC is equal to
80°
SSC CHSL 202176)The difference between the two perpendicular sides of a right-angles triangle is 17 cm and its area is 84cm2. What is the perimeter (in cm) of the triangle?
56
SSC CHSL 202177)The perimeter of an isosceles triangle is 220 cm. If the base is 40 cm, then the length of each of the other sides is:
90 cm
SSC CHSL 202178)If the perimeter of an isosceles right triangle is 15(√2 + 1) cm, then the area of the triangle will be:
56.25 cm2
SSC CHSL 202179)A circle is inscribed in a right-angled triangle. The lengths of the two sides containing the right angle are 15 cm and 8 cm. What is the radius of the in-circle?
3 cm
SSC CHSL 202180)In ΔABC, D and E are the points on the sides AB and AC, respectively such that ∠AED = ∠ABC. If AE = 6 cm, BD = 2 cm, DE = 3 cm and BC = 5 cm, then (AB + AC) is equal to:
\(\frac{70}{3}\) cm
SSC CHSL 202181)The perimeter of a right-angled triangle whose sides that make right angles are 15 cm and 20 cm is:
60 cm
SSC CHSL 202182)Points D, E and F are on the sides AB, BC and AC, respectively, of triangle ABC such that AE, BF and CD bisect ∠A, ∠B and ∠C, respectively. If AB = 6 cm, BC = 7 cm and AC = 8 cm, then what will be the length of BE?
3 cm
SSC CHSL 202183)If the angle between the internal bisectors of two angles ∠B and ∠C of a triangle ABC is 125°, then the value of ∠A is:
70°
SSC CHSL 202184)What is the area (in cm2, correct to one decimal place) of a triangle whose base is 21.4 cm and height is 15.5 cm?
165.9
SSC CHSL 202185)In ΔPQR, points T and S are on PQ and PR, respectively, such that TS is parallel to QR. If TQ = 7.2 cm, PS = 1.8 cm and SR = 5.4 cm, then find the length of PT.
2.4 cm
SSC CHSL 202186)D, E and F are the feet of the perpendiculars from the vertices A, B and C, respectively, of a triangle ABC. If angle BED and angle BFE (in degrees) are 24 and 110, respectively, what is the measure (in degrees) of angle EBF?
46
SSC CHSL 202187)In a triangle ABC, if BD and CD bisect ∠B and ∠C, respectively, and ∠A = 100°, then find ∠BDC.
140°
SSC CHSL 202188)In ΔABC, ∠A = 66°. If 'I' is the incentre of the triangle, then the measure of ∠BIC will be:
123°
SSC CHSL 202189)In ΔXYZ, P is the midpoint of side XZ and Q is a point on side XY such that QZ bisects PY. If XQ = 24 cm, then what is the length (in cm) of QY?
12
SSC CHSL 202190)In an isosceles triangle ABC, AB = AC and AD is perpendicular to BC. If AD = 12 cm and the perimeter of ΔABC is 36 cm, then the length of BC (in cm) is
10
SSC CHSL 202191)In a triangle ABC, ∠BAC = 90° and AD is perpendicular to BC. If AD = 8.4 cm and BD = 4.8 cm, then the length of BC is
19.5 cm
SSC CHSL 202192)In a right angled triangle ABC, if ∠ABC = 90°, AB = 6 cm, BC = 8 cm, and BD is perpendicular to AC, then AD ∶ DC is:
9 ∶ 16
SSC CHSL 202193)In ΔPQR, ∠PQR = 135°, PQ = 8√2 cm and PR = 17 cm. What is the length (in cm) of QR?
7
SSC CHSL 202194)In ΔABC, D and E are points on sides AB and BC, respectively, such that BD ∶ DA = 1 ∶ 2 and CE ∶ EB = 1 ∶ 4, If DC and AE intersect at F, then FD ∶ FC is equal to:
8 ∶ 3
SSC CHSL 202195)ΔPQR is inscribed in a circle. The bisector of ∠P cuts QR at S and the circle at T .If PR = 5 cm, PS = 6 cm and ST = 4 cm, then the length (in cm ) of PQ is:
12
SSC CHSL 202196)In Δ ABC, DE || BC, where D and E are points on the sides AB and AC, respectively. If AD = 2 cm, BD = 5.2 cm, AC = 9 cm and AE = x cm, then what is the value of x?
2.5
SSC CHSL 202197)ΔABC is right-angled at B and D is a point on AC such that BD is perpendicular to AC. If BD = 6 cm and AD = 3 cm, then what will be the length of AC?
15 cm
SSC CHSL 202198)In \( \triangle ABC\), ∠B = 90° . AD and CE are the medians drawn from A and C, respectively. If AC = 10 cm and AD =√ 55 cm, then the length of CE is:
√ 70 cm
SSC CHSL 202199)The side BC of \(\triangle ABC\) is produced to D. The bisectors of ∠ ABC and ∠ ACD meet at E. If AB = AC and ∠ BEC = 35°, then the measures of ∠ABC will be:
55°
SSC CHSL 2021100)The sides AB and AC of a ∆ABC are produced up to points D and E. The bisectors of the exterior angles to formed. intersect each other at point I if ∠ACB is 660 and ∠ABC is 44, then what is the measure (in degrees) of ∠BIC?
55
SSC CHSL 2021101)The side BC of a triangle ABC is extended to a point D. If ∠ACD = 117° and ∠ABC = \frac{5}{8} ∠BAC. then what is the measure of ∠ABC ?
45°
SSC CHSL 2021102)How many isosceles triangles with integer sides are possible such that the sum of two of the sides is 16 cm?
24
SSC CHSL 2021103)In ∆ ABC, ∠B = 900, AB = 8 cm and BC = 15 cm, D is a point on BC such that AD bisects ∠A. The length (in cm) of BD is:
4.8
SSC CHSL 2021104)Two sides of a triangle are 12.8 m and 9.6 m. If the height of the triangle is 12 m, corresponding to 9.6 m then what its height (in m) corresponding to 12.8 m?
9
SSC CHSL 2021105)If ΔABC, DE || AB, where D and E are points on sides AC and BC, respectively. F is a point between C and D such that EF || BD. If AD = 15 cm, DC = 10 cm, then the length of CF is:
4 cm
SSC CHSL 2021106)ΔPQR is inscribed in a circle with center O. PO is produced to meet QR at U and the circle at S, and PT ⊥ QR, where T lies between Q and U. if ∠Q = 70° and ∠R = 55°, then what is the measure (in degrees) of ∠TPS?
15
SSC CHSL 2021107)If a and b are the lengths of two sides of a triangle such that the product ab = 24, where a and b are integers, then how many such triangles are possible?
16
SSC CHSL 2021108)Two circles touch internally. Their diameters are, respectively, 8 cm and 4 cm. What is the distance (in cm) between their centres?
2
SSC CHSL 2021109)ΔPQR is an isosceles triangle with PQ = PR = 25 cm. If PS is the median on QR from P such that PS = 7 cm, then the length of QR is:
48 cm
SSC CHSL 2021110)The centroid of an equilateral triangle ΔABC is O. If AB = 27 cm, then the length of AO is:
9√3 cm
SSC CHSL 2021111)In ΔABC, AD is a median. If points E, F and G are midpoints of AD, AE and DE, respectively, then what will be the area ΔBFG ?
\(\frac{1}{4}\) (Area of ΔABC)
SSC CHSL 2021112)The altitude AD of a triangle ABC is 9 cm. If AB = 6√3 cm and CD = 3√3 cm, then what will be the measure of ∠A?
60°
SSC CHSL 2021113)In ΔABC, ∠A = 135°, CA = 5√2 cm and AB = 7 cm. E and F are midpoints of sides AC and AB, respectively. The length of EF (in cm) is:
6.5
SSC CGL 2020114)In \(\triangle ABC\), \(\angle C = 90^\circ\), AC = 5 cm and BC = 12 cm. The bisector of \( \angle A \) meets BC at D. What is the length of AD ?
\(\frac{5\sqrt{13}}{3}\) cm
By the Pythagoras theorem,
\((AB)^2 = (AC)^2 + (BC)^2\); ⇒ \((AB)^2 = (5)^2 + (12)^2\);
⇒ AB = 13 cm;
By angle bisector theorem,
\(\frac{AB}{BD} = \frac{AC}{CD}\);
Let CD be x cm.
\(\frac{13}{12 - x} = \frac{5}{x};\)
⇒ x = 60/18 = 10/3;
In \(\triangle ACD\),
\((AD)^2 = (AC)^2 + (CD)^2;\)
⇒ \((AD)^2 = (5)^2 + (\frac{10}{3})^2\); ⇒ AD = \(\frac{5\sqrt13}{3}\)
SSC CGL 2020115)D is the midpoint of side BC of \(\triangle ABC\). Point E lies on AC such that \(CE={1\over3}AC\). BE and AD intersect at G. What is \(AG\over GD\) ?
4 : 1
CE = \({AC\over 3} \) and D is the midpoint of BC. Let, M be the midpoint of EC. [\(\therefore DM\parallel BE \) ⇒ \(DM\parallel GE\)]
In \(\triangle ADM\), to apply basic proportionality theorem , \(AE= {2AC\over 3}; \space EC= {AC\over3}\) ; \(EM= {1\over2}\times{AC\over3}= {AC\over6}\) ;
\(\therefore {AG\over GD}= {AE\over EM}= {2AC\over3}\div{AC\over6}\) = 4 : 1
SSC CGL 2020116)In \(\triangle PQR\), PQ = 24 cm. and \(\angle Q = 58^\circ\). S and T are the points on the sides PQ and PR, respectively, such that \(\angle STR = 122^\circ\). If PS = 14 cm and PT = 12 cm, then the length of RT is :
16 cm
\(\angle PTS + \angle STR = 180^0\) ;
\(\angle PTS = 180 - 122 = 58^0\) ;
\(\angle P\) is a common angle.
\(\triangle PQR\) and\( \triangle PTS\) are similar triangle. SO,
\(\frac{PT}{PQ} = \frac{PS}{PR}\) ; ⇒ \(\frac{12}{24} = \frac{14}{PR}\) ; ⇒ PR = 28 cm ;
RT = PR - PT = 28 - 12 = 16 cm
SSC CGL 2020117)In \(\triangle ABC\), \(\angle B=90^0\). If the points D and E are on the side BC such that BD = DE = EC, then which of the following is true?
\(8AE^2 = 3AC^2 + 5AD^2\)
In \(\triangle ABE,\) ⇒ \(AE^2=AB^2+BE^2\) ____(i)
In \(\triangle ABD,\) ⇒ \(AD^2=AB^2+BD^2\) ____(ii)
In \(\triangle ABC,\) ⇒ \(AC^2=AB^2+BC^2\) ____(iii)
\(\therefore AE^2=AB^2+(2BD)^2\) ⇒ \(AE^2=AB^2+4BD^2\) ⇒ \(AE^2=AD^2+3BD^2\) ___(iv)
From equation (i)and (iii), \(BD^2 ={1\over5}(AC^2-AE^2)\) ____(v)
From equation (iv), \(AE^2=AD^2+{3\over5}(AC^2-AE^2)\) ⇒ \(8AE^2=5AD^2+3AC^2\)
SSC CGL 2016118)In ΔABC, ∠B = 70°and ∠C = 60°. The internal bisectors of the two smallest angles of ΔABC meet at O. The angle so formed at O is
125°
SSC CGL 2016119)In a triangle PQR, the side QR is extended to S. ∠QPR = 72° and ∠PRS = 110°, then the value of ∠PQR is:
38°
SSC CGL 2016120)The centroid of a triangle is the point where
the medians meet
SSC CGL 2020121)If angles of a triangle are in the ratio of 2 : 3 : 4, then the measure of the smallest angle is:
\(40^0\)
Let angles of triangle be 2k, 3k and 4k.
\(\therefore 2k+3k+4k=180^0\) ; ⇒ \( k =\) \(20^0\) ;
Value of smallest angle = 2k = \(2\times20^0=40^0\)
SSC CGL 2020122)In the given figure, if \( DE \parallel BC \), AD = 2.5 cm, DB = 3.5 cm and EC = 4.2 cm, then the measure of AC is:
7.2 cm
Let AE = x; \( DE \parallel BC \) \(\therefore \angle ADE= \angle ABC; \space \angle AED = \angle ACB\); By AA - similarity theorem, \(\triangle ADE\sim \triangle ABC\) ; \(\therefore{AD\over AB}={AE\over AC}\) ; ⇒ \({2.5\over2.5+3.5}={x\over x+4.2}\) ; ⇒ x = 3 = AE; \(\therefore\) AC = AE + EC = 3 + 4.2 = 7.2 cm
SSC CGL 2020123)ABC is an equilateral triangle. P, Q and R are the midpoints of sides AB, BC and CA, respectively. If the length of the side of the triangle ABC is 8 cm, then the area of \(\triangle PQR\) is:
\({4\sqrt3}\space cm^2\)
In the\(\triangle ABC\), point P, Q, R are mid points so,
Sides of the \(\triangle PQR\) = 8/2 = 4 cm;
s =\( \frac{perimeter of \triangle PQR}{2} = \frac{4 + 4 + 4}{2} \)= 6 cm;
Area of \(\triangle PQR\) by Heron's formula : \(\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{6(6-4)(6-4)(6-4)}=4\sqrt3\) \(cm^2\)
SSC CGL 2020124)If the area of an equilateral triangle is \(36\sqrt3\space cm^2\), then the perimeter of the triangle is:
36 cm
Area of an equilateral triangle = \(36\sqrt3\space cm^2\); ⇒ \({\sqrt3\over4}\times Side^2= 36\sqrt3\); ⇒ Side = 12 cm ; Therefore, Perimeter of triangle = \(3\times Side\) = 36 cm
SSC CGL 2020125)In the given figure, if AB = 10 cm, \(\angle ABD = 90^0\) and AD = 17 cm, then the measure of CD is:
9 cm
In \(\triangle ABC\), AB = 8 cm ; AC = 10 cm; \(\therefore BC = \sqrt{AC^2-AB^2}=\sqrt{10^2-8^2}=6\space cm\); In \(\triangle ABD\), AD = 17 cm ; \(\therefore BD =\sqrt{AD^2-AB^2}=\sqrt{17^2-8^2}= 15 \space cm\) ; So CD = 15 - 6 = 9 cm
SSC CGL 2020126)In a triangle, if the measures of two sides are 5 cm and 8 cm, then the third side can be:
4 cm
In a triangle, the sum of two sides is greater than the remaining/ third side. 5 + 2 < 8 ; 5 + 3 = 8 ; 5 + 8 < 14 ; But, 5 + 4 > 8 ; So length of third side = 4 cm
SSC CGL 2020127)In a triangle ABC, DE is parallel to BC; AD = a, DB = a + 4, AE = 2a + 3, EC = 7a. What is the value of a if a > 0 ?
3
\(DE\parallel BC\); \(\therefore\angle ADE = \angle ABC\); \(\angle AED = \angle ACB\) ; By AA- similarity theorem, \(\triangle ADE\sim \triangle ABC\); \(\therefore{AD\over AB}={AE\over AC}\) ⇒ \({a\over a+a+4}={2a+3\over 2a+3+7a}\); ⇒ a= 3
SSC CGL 2020128)In \(\triangle ABC\), if the ratio of angles is in the proportion 3 : 5 : 4, then the difference between the biggest and the smallest angles (in degrees) is:
\(30^0\)
We know that sum of the all angles of the triangle is equal to the \(180^0\) .
Let the angles be 3x, 5x and 4x.
so, 3x + 5x + 4x = 180;
x = 15;
Smallest angle = 3x = 3 x 15 = \(45^0\);
Biggest angle = 5x = 5 x 15 = \( 75^0\);
Difference between the biggest and the smallest angles = 75 - 45 = \(30^0\)
SSC CGL 2020129)In the given figure, \(\triangle ABC\) is an isosceles triangle, in which AB = AC, \(AD \perp BC\), BC = 6 cm. and AD = 4 cm. The length of AB is:
5 cm.
AB = AC; \(\angle ADC=\angle ADB=90^0\); \(\therefore BD = DC\) ; BC = 6 cm. ; BD = \(6\over2\)= 3 cm. ; AD = 4 cm. ; In \(\triangle ABD\), \(AB=\sqrt{AD^2+BD^2}=\sqrt{4^2+3^2}\) = 5 cm.
SSC CGL 2020130)In the given figure, the measure of \(\angle BAC\) is:
\(48^0\)
In a triangle exterior angle = sum of other two interior angles ; \(\angle ACD = \angle BAC+\angle ABC;\) ⇒ \(\angle BAC= \angle ACD-\angle ABC\) = \(110^0-62^0= 48^0\)
SSC CGL 2020131)In \(\triangle ABC\), D, E and F, are the midpoints of sides AB, BC and CA, respectively. If AB = 12 cm, BC = 20 cm and CA = 15 cm, then the value of \({1\over2}(DE+EF+DF)\) is:
11.75 cm
By mid point theorem,
DF = BC/2 = 20/2 = 10 cm;
DE = AC/2 = 15/2 = 7.5 cm;
EF = AB/2 = 12/2 = 6 cm;
\({1\over2}(DF + DE+EF)={1\over2}(10+7.5+6) =11.75\)
SSC CGL 2020132)The length of each equal side of an isosceles triangle is 15 cm and the included angle between those two sides is 90°.Find the area of the triangle.
\({225\over2}cm.^2\)
Area of \(\triangle ABC = {1\over2}\times AB \times AC ={1\over2}\times 15\times 15 ={225\over2} cm^2\)
SSC CGL 2020133)In \(\triangle ABC,\) D is a point on BC such that AD is the bisector of \(\angle A\), AB = 11.7 cm. AC = 7.8 cm. and BC = 13 cm. What is the length (in cm.) of DC?
5.2
AD is bisector of \(\angle A\).
\({AB\over AC}={BD\over DC}\); Let, BD = x cm. ; DC = (13 - x)cm. ;
\({11.7\over7.8}= {x\over13-x}\); x = 7.8 cm. ;
DC = 13 - 7.8 = 5.2 cm
SSC CGL 2020134)In the given figure, a circle inscribed in triangle PQR touches its sides PQ, QR and RP at points S, T and U,respectively. If PQ = 15 cm, QR= 10 cm, and RP = 12 cm, then find the lengths of PS, QT and RU?
PS = 8.5 cm, QT = 6.5 cm and RU = 3.5 cm
PQ = 15 cm, QR= 10 cm, and RP = 12 cm. ;
PQ = x + y = 15------------------(i);
QR = y + z = 10----------(ii);
RP = z + x = 12 ------(iii);
Solving equation, we get :
z = 3.5 cm = RU;
y = 6.5 cm = QT;
x = 8.5 cm = PS
SSC CGL 2020135)In the triangle, If AB = AC and \(\angle ABC = 72^0\), then \(\angle BAC\) is:
\(36^0\)
By triangle properties,
When AB = AC then \(\angle ABC = \angle ACB\) so,
In\(\triangle ABC,\)
\(\Rightarrow \angle ABC + \angle ACB + \angle BAC = 180^{0}\);
\(\Rightarrow \angle ABC + \angle ABC + \angle BAC = 180^{0}\);
\(\Rightarrow 72^{0} + 72^{0} + \angle BAC = 180^{0}\);
\(\Rightarrow \angle BAC = 180^{0} - 144^{0} = 36^{0}\)
SSC CGL 2020136)Arrange the angles of the triangle from smallest to largest in the triangle, where the sides are AB = 7 cm, AC = 8 cm, and BC = 9 cm.
C,B,A
Measurement of angle is according to the length of opposite line.
So, The order of the angles of the triangle from smallest to largest in the triangle,
C < B < A.
SSC CGL 2019137)The bisector of \(\angle B\) in \(\triangle ABC\) meets AC at D. If AB = 10 cm, BC = 11 cm and AC = 14 cm, then the length of AD is:
\(20\over3\) cm
SSC CGL 2019138)In \(\triangle ABC\), AB = AC and D is a point on BC. If BD = 5 cm, AB = 12 cm and AD = 8 cm, then the length of CD is:
16 cm
SSC CGL 2019139)The sides PQ and PR of \(\triangle PQR\) are produced to points S and T, respectively. The bisectors of \(\angle SQR\) and \(\angle TRQ\)meet at U. If \(\angle QUR=79^0\), then the measure of \(\angle P\) is:
\(22^0\)
SSC CGL 2019140)In \(\triangle PQR\), I is the incentre of the triangle. If \(\angle QIR=107^0\), then what is the measure of \(\angle P\)?
\(34^0\)
SSC CGL 2019141)The perimeters of two similar triangles ABC and PQR are 78 cm and 46.8 cm, respectively. If PQ = 11.7 cm., then the length of AB is:
19.5 cm.
SSC CGL 2019142)In \(\triangle ABC\), the perpendiculars drawn from A, B and C meet the opposite sides at D, E and F, respectively. AD, BE and CF intersect at point P. If \(\angle EPD=116^0\) and the bisectors of \(\angle A\) and \(\angle B\) meet at Q, then the measure of is \(\angle AQB\) is :
\(122^0\)
SSC CGL 2019143)△ABC and △DBC are on the same BC but on opposite sides of it. AD and BC intersect each other at O.If AO = a cm, DO = b cm and the area of △ABC = x cm sq, then what is the area(in cm sq.) of △DBC ?
\({bx\over a}\)
SSC CGL 2019144)The sides of a triangle are 56 cm, 90 cm and 106 cm. The circumference of its circumcircle is:
\(106\pi\)
SSC CGL 2019145)In \(\triangle ABC\), the medians AD, BE and CF meet at O. What is the ratio of the area of \(\triangle ABD\) to the area of \(\triangle AOE\)?
3 : 1
SSC CGL 2019146)In \(\triangle ABD\), C is the midpoint of BD. If AB = 10 cm, AD = 12 cm and AC = 9 cm, then BD = ?
\(2\sqrt {41}\) cm.
SSC CGL 2019147)The bisector of \(\angle A\) in \(\triangle ABC\) meets BC in D. If AB = 15cm, AC = 13cm and BC = 14cm,then DC = ?
6.5 cm
SSC CGL 2019148)If in \(\triangle PQR\), \(\angle P = 120^0\), \(PS \perp QR\) at S and PQ + QS = SR, then the measure of \(\angle Q\) is :
\(40^0\)
SSC CGL 2019149)In \(\triangle ABC\), D and E are the points on AB and AC respectively such that \(AD\times AC = \)\(AB\times AE\). If \(\angle ADE= \angle ACB+30^0\) and \(\angle ABC= 78^0\), then \(\angle A = ?\)
\(54^0\)
SSC CGL 2019150)In \(\triangle PQR \), \(\angle Q> \) \(\angle R\), PS is the bisector of \(\angle P\) and \(PT \perp QR\), If \(\angle SPT = 28^0\) and \(\angle R = 23^0\), then the measure of \(\angle Q\) is :
\(79^0\)
SSC CGL 2019151)In \(\triangle ABC\), \(BE \perp AC\), \(CD \perp AB\) and BE and CD intersect each other at O. The bisectors of \(\angle OBC\) and \(\angle OCB\) meet at P. If \(\angle BPC = 148^0\), then what is the measure of \(\angle A\) ?
\(64^0\)
SSC CGL 2019152)If in \(\triangle ABC\), D and E are the points on AB and BC respectively such that \(DE \parallel AC \), and AD : AB = 3 : 8, then (area of \(\triangle BDE\) ) : ( area of quadrilateral DECA) = ?
25 : 39
Let AB = 8 unit and AD = 3 unit, thus BD = 8 - 3 = 5 unit
since DE || AC therefore ΔBDE~ΔBAC
We know that if the triangles are similar, the ratio of areas of the triangles is equal to the square of their sides
Area of ΔBDE/area of ΔBAC = (BD/AB)2
Area of ΔBDE/area of ΔBAC = (5/8)2 = 25/64
Let area of ΔBDE = 25 unit and area of ΔBAC = 64 unit
Area of quadrilateral DECA = area of ΔBAC – area of ΔBDE = 64 – 25 = 39 unit
Area of ΔBDE : Area of quadrilateral DECA = 25 : 39
SSC CGL 2019153)S is the incenter of \(\triangle PQR\) . If \(\angle PSR = 125^0\) , then the measure of \(\angle PQR\) is:
\(70^0\)
SSC CGL 2019154)The sides of a triangle are 12 cm, 35 cm and 37 cm. What is the circumradius of the traingle?
18.5 cm.
SSC CGL 2019155)A circle is inscribed in \( \triangle ABC\) , touching AB, BC and AC at the points P, Q and R respectively. If AB - BC = 4 cm, AB - AC = 2 cm and the perimeter of \(\triangle ABC\) = 32 cm, then PB + AR is equal to:
\(38\over3\) cm
Perimeter = 32 cm;
AB + BC + AC = 32 cm ---(1);
AB - BC = 4 cm ---(2);
AB - AC = 2 cm ---(3);
On eq(1) + (2) + (3),
3AB = 38;
AB = PB + AR = 38/3
SSC CGL 2019156)In\( \triangle ABC\),\( \angle A = 58^\circ.\) If I is the in center of the triangle, then the measure of \(\angle BIC\) is:
\(119^0\)
\(\angle BIC = 90^0 + \angle A/2\);
\(\angle BIC = 90^0+ 58/2 = 90^0 + 29^0 = 119^0\)
SSC CGL 2019157)The sides of a triangle are 11 cm, 60 cm and 61 cm. What is the radius of the circle circumscribing the triangle?
30.5 cm
Radius of the circle circumference = hypotenuse/2 = 61/2 = 30.5 cm
SSC CGL 2019158)In \( \triangle ABC\), AB=7cm, BC=10cm, and AC = 8 cm. If AD is the angle bisector of \(\angle BAC\), where D is a point on BC, then BD is equal to:
\({14 \over3}cm\)
Let the BD = x,
DC = 10 - x
By angle bisector theorem,
\(\frac{BD}{CD}\) =\( \frac{AB}{AC} \); \(\frac{x}{10 - x}\) = \(\frac{7}{8}\)
8x = 70 - 7x
15x = 70
x = 14/3 cm
BD = \(\frac{14}{3}\)
SSC CGL 2019159)In \(\Delta ABC\), AB = 6 cm, AC = 8 cm, and BC = 9 cm. The length of median AD is :
\({\sqrt{119}\over 2}\) cm
In ΔABC, AD is median . D. will be the mid point of BC.
BD = CD = BC/2= 9/2 = 4.5 cm;
\(AB^2 + AC^2 = 2(AD^2 + BD^2);\)
\(6^2 + 8^2 = 2(AD^2 + (4.5)^2);\)
\(100/2 = AD^2 + 20.25\);
\(AD^2 = 29.75 = 119/4;\)
\(AD = \frac{\sqrt{119}}{2}\)
SSC CGL 2019160)In \({\Delta ABC}, \angle A =52^0\) and O is the orthocentre of the triangle (BO and CO meet AC and AB at E and F respectively when produced). If the bisectors of \({\angle OBC} \) and \({\angle OCB}\) meet at P, then the measure of \({\angle BPC}\) is :
154º
By the orthogonal property,
\(\angle BOC = 180 - \angle A\);
\(\angle OBC + \angle OCB = \angle A;\)
\(\angle BPC = 180 - \angle A/2;\)
\(\angle BPC = 180 - \angle 52/2 = 180 - 26 = 154^0\)
SSC CGL 2019161)In \( {\Delta ABC}\), D is a point on side BC such that \({\angle ADC}={\angle BAC}\). If CA = 12cm, CB = 8cm, then CD is equal to :
18 cm.
\(\angle ADC = \angle BAC; \angle ACB = \angle ACD; So, \triangle ABC ~ \triangle DCA,; So, \frac{BC}{AC} = \frac{AC}{CD}; \frac{8}{12} = \frac{12}{CD};\)
SSC CGL 2019162)The sides AB and AC of \( {\Delta ABC}\) are produced to P and Q resepectively. The bisectors of \({\angle CBP}\) and\({\angle BCQ}\) meet at R. If the measure of \({\angle A}\) is 44º, then what is the measure of \({1\over2 }\)\({\angle BRC}\)?
34º
\(\angle BRC = 90 ^o - \angle A/2; = 90 ^o - 44^o/2 = 90 ^o - 22^o = 68^o; \frac{1}{2} \angle BOC = \frac{68^o}{2}; \frac{1}{2} \angle BOC = 34^o;\)
SSC CGL 2020163)In the given fig. triangle ABC, \(\theta = 80^o\) the measure of each of the other two angle will be:
\(50^0\)
since the two sides are equal in length, and equal side corresponds equal angle, therefore \(2x + \theta = 180\), x = 50