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Quant: Mar '25 to Apr '25

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CAT 2025 Lesson : Quant: Mar '25 to Apr '25 - Lines & Triangles - 07 Apr 2025

Triangles – Past Papers

1) If

a, b, c

are the sides of a triangle, and

a^{2} + b^{2} + c^{2} = bc + ca + ab

, then the triangle is

(1) equilateral
(2) isosceles
(3) right angled
(4) obtuse angled

2)

In the figure above, AB = BC = CD = DE = EF = FG = GA. Then ∠DAE is approximately

(1) 15

\degree

(2) 20

\degree

(3) 30

\degree

(4) 25

\degree

3) On a straight road XY, 100 metres in length, 5 stones are kept beginning from the end X. The distance between two adjacent stones is 2 metres. A man is asked to collect the stones one at a time and put at the end Y. What is the distance covered by him?

(1) 460 metres
(2) 540 metres
(3) 860 metres
(4) 920 metres

4) The internal bisector of an angle A in a triangle ABC meets the side BC at point D. AB =

4

, AC =

3

and ∠A =

60\degree

. Then what is the length of the bisector AD?

(1)

\dfrac{12 \sqrt{3}}{7}

(2)

\dfrac{12 \sqrt{13}}{7}

(3)

\dfrac{4 \sqrt{3}}{7}

(4)

\dfrac{4 \sqrt{3}}{7}

5) In the figure below, ABC is a right-angled triangle. AD is the altitude. Circles are inscribed within the triangles ACD and ABD. P and Q are the centres of the circles. The distance PQ is (1) 7 m
(2) 4.5 m
(3) 10.5 m
(4) 6 m

Choose the option that correctly states the data required to answer the question below

6) D, E, F are the mid-points of the sides AB, BC and CA of triangle ABC respectively. What is the area of DEF in square centimetres?
A. AD = 1 cm, DF = 1 cm and perimeter of DEF = 3 cm
B. Perimeter of ABC = 6 cm, AB = 2 cm, and AC = 2 cm

(1) Statement A alone is sufficient
(2) Statement B alone is sufficient
(3) Both Statements A and B together are sufficient
(4) Neither Statement is sufficient

7) A vertical tower OP stands at the centre O of a square ABCD. Let h and b denote the lengths OP and AB respectively. Suppose ∠APB = 60

\degree

, then the relationship between h and b can be expressed as

(1) 2b

^2

= h

^2

(2) 2h

^2

= b

^2

(3) 3b

^2

= 2h

^2

(4) 3h

^2

= 2b

^2

8) In the diagram given below,
∠ABD = ∠CDB = ∠PQD = 90

\degree

. If AB : CD = 3 : 1, the ratio of CD : PQ is

(1) 1 : 0.69
(2) 1 : 0.75
(3) 1 : 0.72
(4) None of the above

9) In the figure (not drawn to scale) given below, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC and QD is parallel to CP. In ΔARC, ∠ARC = 90

\degree

, and in ΔPQS, ∠PSQ = 90

\degree

. The length of QS is 6 cm. What is the ratio AP : PD?

(1) 10 : 3
(2) 2 : 1
(3) 7 : 3
(4) 8 : 3

10) Consider the triangle ABC shown in the following figure where BC = 12 cm, DB = 9 cm, CD = 6 cm and ∠BCD = ∠BAC. What is the ratio of the perimeter of the triangle ADC to that of the triangle BDC?

(1) 7/9
(2) 8/9
(3) 6/9
(4) 5/9

11) An equilateral triangle BPC is drawn inside a square ABCD. What is the value of the angle APD in degrees?

(1) 75
(2) 90
(3) 120
(4) 135
(5) 150

12) In a triangle ABC, the lengths of the sides AB and AC equal 17.5 cm and 9 cm respectively. Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD = 3 cm, then what is the radius (in cm) of the circle circumscribing the triangle ABC?

(1) 17.05
(2) 27.85
(3) 22.45
(4) 32.25
(5) 26.25

13) Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is

Answer:

14) A triangle ABC has area 32 sq units and its side BC, of length 8 units, lies on the line x = 4. Then the shortest possible distance between A and the point (0, 0) is

(1) 8 units
(2) 4 units
(3)

4 \sqrt{2}

units
(4)

2 \sqrt{2}

units

15) The sum of the perimeters of an equilateral triangle and a rectangle is 90cm. The area, T, of the triangle and the area, R, of the rectangle both in sq cm satisfy the relationship R = T

^2

. If the sides of the rectangle are in the ratio 1:3, then the length,in cm, of the rectangle, is

(1) 21
(2) 24
(3) 27
(4) 18

16) A Circle of diameter 8 inches is inscribed in a triangle ABC where
Answer:

17) Let D and E be points on sides AB and AC, respectively, of a triangle ABC, such that AD : BD = 2 : 1 and AE : CE = 2 : 3. If the area of the triangle ADE is 8 sq cm, then the area of the triangle ABC, in sq cm, is

Answer:

Solution

1) (1) equilateral
2) (4) 25

\degree

3) (3) 860 metres
4) (1)

\dfrac{12 \sqrt{3}}{7}

5) (1) 7 m
6) (2) Statement B alone is sufficient
7) (2)

2 h^2 = b^2

8) (2) 1 : 0.75
9) (3) 7 : 3
10) (1) 7/9
11) (5) 150
12) (5) 26.25
13) 24
14) (2) 4 units
15) (3) 27
16) 120
17) 30

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