TRIANGLE

level – 1

1. If the area of a triangle is 864 cm² and the ratio of its base to the corresponding altitude is 4 : 3, then the altitude of the triangle is:
a) 18 cm
b) 24 cm
c) 27 cm
d) 36 cm

2. The base of a triangle is 18 cm and its height is 10 cm. The height of another triangle of triple the area, having the base 15 cm, is:
a) 18 cm
b) 20 cm
c) 24 cm
d) 30 cm

3. The sides of a triangle are 6 cm, 8 cm, and 10 cm. The area (in cm²) of the triangle formed by joining the midpoints of its sides is:
a) 6
b) 12
c) 18
d) 24

4. The sides of a triangle are in the ratio 3 : 4 : 5. If the perimeter of the triangle is 24 cm, then the area (in cm²) of the triangle is:
a) 24
b) 36
c) 48
d) 60

5. The measures (in cm) of the sides of a right-angled triangle are given by three consecutive even integers. The area (in cm²) of the triangle is:
a) 24
b) 30
c) 36
d) 48

6. The lengths of the three medians of a triangle are 6 cm, 8 cm, and 10 cm. The area (in cm²) of the triangle is:
a) 24
b) 32
c) 40
d) 48

7. The ratio of the sides of a triangle is 3 : 4 : 5. If the area of the triangle is 96 square units, then the length of the smallest side is:
a) 9 units
b) 12 units
c) 15 units
d) 18 units

8. What is the area of a triangle having perimeter 30 cm, one side 10 cm, and the difference of the other two sides 2 cm?
a) 12 cm²
b) 16 cm²
c) 24 cm²
d) 30 cm²

9. The sides of a triangle are 9 cm, 12 cm, and 15 cm. Find the area of the triangle.
a) 36 cm²
b) 45 cm²
c) 54 cm²
d) 60 cm²

10. Two triangles △PQR and △XYZ are similar. If PQ = 12 cm and XY = 9 cm, then the ratio of the areas of triangles PQR and XYZ is:
a) 3 : 2
b) 4 : 3
c) 16 : 9
d) 9 : 16

11. If △PQR ∼ △XYZ such that QR = 4 cm, YZ = 6 cm and the area of △PQR is 64 cm², then the area of △XYZ is:
a) 96 cm²
b) 120 cm²
c) 144 cm²
d) 192 cm²

12. The areas of two similar triangles △PQR and △XYZ are 36 cm² and 64 cm² respectively. If PQ = 6 cm, then the length of XY is:
a) 8 cm
b) 10 cm
c) 12 cm
d) 16 cm

13. In triangle ABC a line drawn parallel to base BC intersects AB and AC at points D and E respectively. If the area of △ABE is 25 cm², then the area of △ACD is:
a) 20 cm²
b) 25 cm²
c) 30 cm²
d) 50 cm²

14. The perimeter of a triangle is 24 m and its sides are in the ratio 3 : 4 : 5. The area of the triangle is:
a) 18 m²
b) 24 m²
c) 30 m²
d) 36 m²

15. If x and y are the lengths of the perpendicular sides of a right-angled triangle whose hypotenuse is 13 cm and whose area is 30 cm², then the value of (x + y)² is:
a) 169
b) 289
c) 361
d) 400

16. If the ratio of the altitudes of two triangles is 2 : 3 and the ratio of their corresponding areas is 8 : 9, then the ratio of their corresponding lengths of bases is:
a) 3 : 4
b) 4 : 3
c) 4 : 5
d) 6 : 5

17. The in-radius of a triangle is 5 cm, and the sum of the lengths of its sides is 36 cm. The area of the triangle (in square cm) is:
a) 80
b) 85
c) 90
d) 100

18. Two triangles △PQR and △XYZ are similar. If the ratio of their perimeters is 3 : 2, then the ratio of their areas is:
a) 3 : 2
b) 9 : 4
c) 6 : 5
d) 4 : 3

19. The hypotenuse of a right-angled triangle is 25 cm and the difference of the other two sides is 7 cm. Then, the area of the triangle is:
a) 60 cm²
b) 84 cm²
c) 96 cm²
d) 120 cm²

20. △PQR ∼ △XYZ. If the area of △PQR is 16 cm² and the area of △XYZ is 36 cm², and QR = 4 cm, then the length of YZ is:
a) 5 cm
b) 6 cm
c) 8 cm
d) 9 cm

21. In △PQR, points X and Y are the midpoints of PQ and PR respectively. The ratio of the areas of △PXY and quadrilateral QRYX is:
a) 1 : 2
b) 1 : 3
c) 1 : 4
d) 2 : 3

22. A string of length 40 cm is bent first into a square and then into a right-angled triangle, keeping one side of the square fixed as its base. The area of the triangle (in cm²) is:
a) 40
b) 48
c) 60
d) 80

23. The centroid of a triangle △PQR is G. If the area of △PQR is 90 sq. units, then the area of △QGR is:
a) 20
b) 25
c) 30
d) 45

24. In a triangle PQR, PQ = 9 cm, PR = 15 cm and ∠Q = 90°. Then the area of △PQR is:
a) 36 cm²
b) 45 cm²
c) 54 cm²
d) 60 cm²

25. In a triangle ABC, a line DE is drawn parallel to BC intersecting AB and AC at D and E respectively. If DE = 4 cm, BC = 8 cm and the area of △ADE = 20 sq. cm, then the area of △ABC is:
a) 40 sq. cm
b) 60 sq. cm
c) 80 sq. cm
d) 100 sq. cm

26. △PQR is similar to △XYZ. If the ratio of their corresponding sides is 3 : 2, then the ratio of their areas is:
a) 3 : 2
b) 9 : 4
c) 6 : 4
d) 3 : 4

27. The inradius of a triangle is 5 cm and its area is 60 sq. cm. The perimeter of the triangle is:
a) 20 cm
b) 22 cm
c) 24 cm
d) 26 cm

28. In △PQR, the medians PX and QY intersect at G. Find the ratio of the areas of △PGQ and the quadrilateral GXRY:
a) 1 : 2
b) 1 : 3
c) 2 : 3
d) 1 : 5

29. P, Q and R are the midpoints of the sides XY, YZ and ZX respectively of a triangle △XYZ. Then the ratio of the areas of △PQR and △XYZ is:
a) 1 : 2
b) 1 : 3
c) 1 : 4
d) 3 : 4

level – 2

1. The radius of the incircle of a triangle is 3 cm. If the area of the triangle is 24 cm², then its perimeter is:
a) 12 cm
b) 14 cm
c) 16 cm
d) 18 cm

2. The perimeter of a right-angled triangle is 60 cm and its hypotenuse is 26 cm. Find the area of the triangle:
a) 100 cm²
b) 110 cm²
c) 120 cm²
d) 130 cm²

3. Two triangles have the same base of 14 cm. Their heights are 8 cm and 6 cm respectively. Find the ratio of their areas:
a) 3 : 4
b) 4 : 3
c) 2 : 3
d) 3 : 2

4. A parallelogram and a triangle share the same base of 18 cm and have equal areas. If the height of the parallelogram is 10 cm, find the height of the triangle:
a) 16 cm
b) 18 cm
c) 20 cm
d) 22 cm

5. In △ABC, EF is drawn parallel to BC where E is on AB and F is on AC such that AE : AB = 2 : 5. If the area of △ABC is 100 sq. cm, find the area of △AEF:
a) 14 cm²
b) 15 cm²
c) 16 cm²
d) 18 cm²

6. In a right-angled triangle with legs 15 cm and 20 cm, find the length of the altitude drawn to the hypotenuse and the two segments it creates:
a) Altitude = 12, segments = 9 and 16
b) Altitude = 10, segments = 8 and 17
c) Altitude = 12, segments = 8 and 17
d) Altitude = 10, segments = 9 and 16

7. The centroid of △PQR is G and PX is a median. If the area of △PQR is 42 sq. cm, find the area of △PGQ:
a) 12 cm²
b) 13 cm²
c) 14 cm²
d) 15 cm²

8. An isosceles triangle has equal sides of 13 cm and a base of 10 cm. Find its area:
a) 55 cm²
b) 58 cm²
c) 60 cm²
d) 64 cm²

9. Two similar triangles have corresponding sides in the ratio 2 : 3. If the area of the smaller triangle is 200 sq. cm, find the area of the larger triangle:
a) 400 cm²
b) 420 cm²
c) 440 cm²
d) 450 cm²

10. The base of a triangle is increased by 20% and its corresponding height is decreased by 20%. What is the percentage change in its area:
a) 2% decrease
b) 4% decrease
c) 4% increase
d) No change

level – 3

1. The sides of a triangular park are 9 m, 12 m and 15 m respectively. Find the cost of grassing the park at the rate of ₹4 per sq. m:
a) ₹180
b) ₹216
c) ₹240
d) ₹300

2. The inradius of a right-angled triangle is 3 cm and its hypotenuse is 13 cm. Find the area of the triangle:
a) 42 cm²
b) 44 cm²
c) 46 cm²
d) 48 cm²

3. The perimeters of two similar triangles are in the ratio 4 : 7. If the area of the smaller triangle is 48 sq. cm, find the area of the larger triangle:
a) 144 cm²
b) 147 cm²
c) 149 cm²
d) 150 cm²

4. In a right-angled triangle, the legs are 15 cm and 20 cm. Find the length of the altitude drawn to the hypotenuse:
a) 10 cm
b) 11 cm
c) 12 cm
d) 13 cm

5. The sides of an isosceles triangle are 10 cm, 10 cm and 12 cm. Find the perimeter of the triangle formed by joining the midpoints of its sides:
a) 12 cm
b) 14 cm
c) 16 cm
d) 18 cm

6. In △ABC, D is the midpoint of AB and E is the midpoint of BC. If the area of △ABC is 120 sq. cm, find the area of △BDE:
a) 24 cm²
b) 28 cm²
c) 30 cm²
d) 36 cm²

7. In a right-angled triangle, the altitude drawn to the hypotenuse is 12 cm. If one of the legs is 15 cm, find the other leg:
a) 18 cm
b) 19 cm
c) 20 cm
d) 21 cm

8. △PQR ∼ △DEF. The perimeter of △PQR is 40 cm and the area ratio of △PQR to △DEF is 25 : 36. Find the perimeter of △DEF:
a) 44 cm
b) 46 cm
c) 48 cm
d) 50 cm

9. In a right-angled triangle, the median drawn to the hypotenuse is 10 cm and the area of the triangle is 96 sq. cm. Find the legs of the triangle:
a) 8 cm and 14 cm
b) 10 cm and 12 cm
c) 12 cm and 16 cm
d) 14 cm and 18 cm

10. The sides of a triangle are 10 cm, 10 cm and 12 cm. Using Heron’s formula verify the area and find how many such triangles can tile a rectangle of dimensions 48 cm × 16 cm:
a) 16
b) 18
c) 20
d) 24

ANSWERS

Isosceles Triangle

level – 1

1. In an isosceles triangle, each of the equal sides is 13 cm and the angle between them is 60°. The area of the triangle is:
a) 52 cm²
b) 56 cm²
c) 65 cm²
d) 78 cm²

2. The area of the triangle formed by the straight line 4x + 3y = 12 and the coordinate axes is:
a) 6 sq. units
b) 9 sq. units
c) 12 sq. units
d) 18 sq. units

3. The area of an isosceles triangle is 12 square units. If the length of the base is 6 units, then the length of each equal side is:
a) 4 units
b) 5 units
c) 6 units
d) 7 units

4. A right-angled isosceles triangle is inscribed in a semi-circle of radius 14 cm such that the hypotenuse of the triangle is the diameter of the semi-circle. The area enclosed by the semi-circle but exterior to the triangle is:
a) 154 cm²
b) 112 cm²
c) 462 cm²
d) 616 cm²

5. The perimeter of an isosceles triangle is 300 cm. Each of the equal sides is 3/4 times the base. The area of the triangle (in cm²) is:
a) 3600
b) 4320
c) 4800
d) 5400

6. The altitude drawn to the base of an isosceles triangle is 6 cm and its perimeter is 48 cm. The area (in cm²) of the triangle is:
a) 48
b) 54
c) 60
d) 72

7. If the length of each of the two equal sides of an isosceles triangle is 13 cm and the adjacent (included) angle is 60°, then the area of the triangle is:
a) 52 cm²
b) 65 cm²
c) 78 cm²
d) 91 cm²

8. Two isosceles triangles have equal vertical angles. If the ratio of their areas is 4 : 25, then the ratio of their corresponding heights is:
a) 2 : 5
b) 4 : 5
c) 5 : 4
d) 16 : 25

9. The centroid of a triangle △PQR is G. If the area of △PQR is 90 cm², then the area of △GRQ is:
a) 20 cm²
b) 25 cm²
c) 30 cm²
d) 45 cm²

10. The ratio of the altitudes of two triangles is 3 : 4 and the ratio of their areas is 9 : 8. The ratio of their corresponding bases is:
a) 3 : 2
b) 4 : 3
c) 8 : 9
d) 9 : 8

level – 2

1. The perimeter of a certain isosceles right-angled triangle is 24 + 12√2 cm. What is the length of the hypotenuse of the triangle?
a) 12 cm
b) 18 cm
c) 24 cm
d) 30 cm

2. In an isosceles triangle, the length of each equal side is 3 times the length of the third side. The ratio of the areas of this isosceles triangle and an equilateral triangle having the same perimeter is:
a) 2 : 1
b) 3 : 2
c) 4 : 3
d) 5 : 4

3. The lengths of two sides of an isosceles triangle are 10 cm and 18 cm respectively. What are the possible values of the perimeter of the triangle?
a) 38 cm and 46 cm
b) 38 cm and 48 cm
c) 36 cm and 46 cm
d) 40 cm and 46 cm

4. The ratio of the areas of two isosceles triangles having the same vertical angle (i.e., angle between the equal sides) is 9 : 16. The ratio of their heights is:
a) 3 : 4
b) 4 : 3
c) 9 : 16
d) 16 : 9

5. The area of an isosceles right-angled triangle is 72 sq. cm. Find the length of its legs and the perimeter of the triangle.
a) Legs = 10 cm, P = 20 + 10√2
b) Legs = 11 cm, P = 22 + 11√2
c) Legs = 12 cm, P = 24 + 12√2
d) Legs = 13 cm, P = 26 + 13√2

6. The lengths of two sides of an isosceles triangle are 8 cm and 15 cm. What are the two possible values of its perimeter? (Check triangle inequality in both cases)
a) 28 cm and 38 cm
b) 30 cm and 38 cm
c) 31 cm and 38 cm
d) 32 cm and 40 cm

7. Two isosceles triangles have the same vertical angle. Their areas are in the ratio 25 : 36. Find the ratio of their corresponding heights.
a) 4 : 5
b) 5 : 6
c) 6 : 7
d) 7 : 8

8. Two isosceles triangles T1 and T2 have equal sides 13 cm, 13 cm, 10 cm and 10 cm, 10 cm, 12 cm respectively. Find the ratio of their areas.
a) 4 : 5
b) 5 : 4
c) 3 : 4
d) 4 : 3

9. The perimeter of an isosceles triangle is 36 cm and its equal sides are 13 cm each. Find the area of the triangle.
a) 54 cm²
b) 58 cm²
c) 60 cm²
d) 64 cm²

10. Two isosceles triangles have the same base of 16 cm. The equal sides of the first triangle are 10 cm each and the equal sides of the second triangle are 17 cm each. Find the ratio of their areas.
a) 2 : 5
b) 3 : 5
c) 4 : 5
d) 2 : 3

ANSWERS

Equilateral Triangle

level – 1

1. From a point inside an equilateral triangle, the perpendiculars to its three sides are 4 cm, 5 cm and 7 cm. The area of the triangle is:
a) 48 cm²
b) 54 cm²
c) 60 cm²
d) 72 cm²

2. The areas of two equilateral triangles are in the ratio 16 : 49. The ratio of their altitudes is:
a) 4 : 7
b) 7 : 4
c) 16 : 49
d) 8 : 14

3. XYZ is an equilateral triangle of side 4 cm. With X, Y, Z as centres and radius 2 cm, three arcs are drawn inside the triangle. The area of the region is:
a) 4π − 4√3 cm²
b) 8π − 8√3 cm²
c) 12π − 8√3 cm²
d) 16π − 12√3 cm²

4. If the numerical value of the perimeter of an equilateral triangle is equal to twice the area, then the side is:
a) 2 units
b) 3 units
c) 4 units
d) 6 units

5. Each side of an equilateral triangle is 8 cm. The area is:
a) 16√3 cm²
b) 24√3 cm²
c) 32√3 cm²
d) 48√3 cm²

6. If the side is increased by 2 units, area increases by 5√3. Original side is:
a) 3 units
b) 4 units
c) 5 units
d) 6 units

7. The height of an equilateral triangle is 12 cm. The area is:
a) 48√3 cm²
b) 72√3 cm²
c) 96√3 cm²
d) 108√3 cm²

8. The area is 12√3 m². The median length is:
a) 4
b) 5
c) 6
d) 8

9. Side = 6 cm. Area between circumcircle and incircle (π = 22/7):
a) 44 cm²
b) 55 cm²
c) 66 cm²
d) 77 cm²

10. The perpendiculars are 4 cm, 5 cm, 7 cm. The area is:
a) 48 cm²
b) 60 cm²
c) 72 cm²
d) 84 cm²

11. The side of an equilateral triangle is 14 cm. Find the area (in cm²) of its incircle:
a) 44
b) 88
c) 154
d) 308

12. The side of an equilateral triangle is 12 cm. The ratio of the areas of its circumcircle to incircle is:
a) 3 : 1
b) 4 : 1
c) 9 : 1
d) 16 : 1

13. If the difference between the areas of the circumcircle and the incircle of an equilateral triangle is 88 cm², then the area of the triangle is (Take π = 22/7):
a) 36√3 cm²
b) 48√3 cm²
c) 64√3 cm²
d) 72√3 cm²

14. A circle is inscribed in an equilateral triangle. Another equilateral triangle is inscribed in the same circle. If the side of the larger triangle is 12 cm, then the ratio of their areas is:
a) 3 : 1
b) 4 : 1
c) 9 : 1
d) 12 : 1

15. The ratio of the areas of the circumcircle and the incircle of an equilateral triangle is:
a) 2 : 1
b) 3 : 1
c) 4 : 1
d) 9 : 1

16. An equilateral triangle is drawn on the side of a square. The ratio of the area of the triangle to the square is:
a) 3 : 4
b) 4 : 3
c) 1 : 2
d) 2 : 1

17. The median of an equilateral triangle is 12 cm. The area is:
a) 48√3
b) 72√3
c) 96√3
d) 144√3

18. If the side is increased by 2 m, area increases by 12√3. Original side is:
a) 4 m
b) 6 m
c) 8 m
d) 10 m

19. If the altitude of an equilateral triangle is 9√3 cm, then the area is:
a) 108 cm²
b) 81 cm²
c) 72 cm²
d) 54 cm²

20. The area of an equilateral triangle inscribed in a circle is 12√3 cm². The area of the circle is:
a) 48 cm²
b) 36 cm²
c) 64 cm²
d) 72 cm²

21. A circle is inscribed in an equilateral triangle of side 12 cm. The area between the triangle and the circle is:
a) 36 cm²
b) 24 cm²
c) 48 cm²
d) 30 cm²

level – 2

1. The area of an equilateral triangle is 144√3 sq. m. Its perimeter is:
a) 36 m
b) 48 m
c) 60 m
d) 72 m

2. From a point in the interior of an equilateral triangle, the perpendicular distances to the three sides are 2 cm, 4 cm and 6 cm. The perimeter (in cm) of the triangle is:
a) 18
b) 24
c) 36
d) 42

3. If the length of each median of an equilateral triangle is 8√3 cm, then the perimeter of the triangle is:
a) 24 cm
b) 36 cm
c) 48 cm
d) 60 cm

4. The area of the circumcircle of an equilateral triangle is 12π sq. cm. The perimeter of the triangle is:
a) 18 cm
b) 24 cm
c) 36 cm
d) 48 cm

5. The sides of an equilateral triangle are increased by 10%, 20%, and 30% respectively. The increase in the perimeter is:
a) 25%
b) 20%
c) 30%
d) 35%

6. If the perimeter of an equilateral triangle is 24 cm, then the length of each median is:
a) 4√3 cm
b) 6√3 cm
c) 8√3 cm
d) 12√3 cm

7. The inradius of an equilateral triangle is 2 cm. The perimeter is:
a) 18 cm
b) 24 cm
c) 12 cm
d) 30 cm

8. The radius of the incircle of an equilateral triangle of side 6 units is x cm. The value of x is:
a) 1
b) 2
c) 3
d) 4

9. The length of a side of an equilateral triangle is 14 cm. Find the area between circumcircle and incircle (π = 22/7):
a) 132 cm²
b) 154 cm²
c) 176 cm²
d) 196 cm²

10. An equilateral triangle has a side of 28 cm. Find the area between circumcircle and incircle (π = 22/7):
a) 528 cm²
b) 572 cm²
c) 616 cm²
d) 660 cm²

Equilateral Triangle Answers