10 Math Worksheet CH-6 Triangles
10 Math Worksheet CH-6 Triangles
10 Math Worksheet CH-6 Triangles
(a) 𝛥𝑃𝑄𝑅 ~ 𝛥𝐶𝐴𝐵 (b) 𝛥𝑃𝑄𝑅 ~ 𝛥𝐴𝐵𝐶 (c) 𝛥𝐶𝐵𝐴 ~ 𝛥𝑃𝑄𝑅 (d) 𝛥𝐵𝐶𝐴 ~ 𝛥𝑃𝑄𝑅
5. In the given figure, two line segments 𝐴𝐶 and 𝐵𝐷 intersect each other at 𝑃
such that 𝑃𝐴 = 6𝑐𝑚, 𝑃𝐵 = 3 𝑐𝑚, 𝑃𝐶 = 2.5 𝑐𝑚, 𝑃𝐷 = 5 𝑐𝑚, ∠𝐴𝑃𝐵 = 50°
and ∠𝐶𝐷𝑃 = 30°, then ∠𝑃𝐵𝐴 is equal to
(a) 50° (b) 30° (c) 60° (d) 100°
6. If in two triangles 𝐷𝐸𝐹 and 𝑃𝑄𝑅, ∠𝐷 = ∠𝑄 and ∠𝑅 = ∠𝐸, then which of
the following is not true?
𝐸𝐹 𝐷𝐹 𝐷𝐸 𝐸𝐹 𝐷𝐸 𝐷𝐹 𝐸𝐹 𝐷𝐸
(a) = (b) = (c) = (d) =
𝑃𝑅 𝑃𝑄 𝑃𝑄 𝑅𝑃 𝑄𝑅 𝑃𝑄 𝑅𝑃 𝑄𝑅
7. In Δ𝐴𝐵𝐶 and Δ𝐷𝐸𝐹, ∠𝐵 = ∠𝐸, ∠𝐹 = ∠𝐶 and 𝐴𝐵 = 3𝐷𝐸. Then, the two triangles are
(a) congruent but not similar (b) similar but not congruent
(c) neither congruent nor similar (d) congruent as well as similar
𝐵𝐶 1 𝑎𝑟(Δ𝑃𝑅𝑄)
8. It is given that Δ𝐴𝐵𝐶 ~ Δ𝑃𝑄𝑅, with 𝑄𝑅 = 3
. Then 𝑎𝑟(Δ𝐵𝐶𝐴) is equal to
1 1
(a) 9 (b) 3 (c) (d)
3 9
1. Is the triangle with sides 25 cm, 5 cm and 24 cm a right triangle? Give reasons for your answer.
2. It is given that DEF ~ RPQ. Is it true to say that D = R and F = P? Why?
3. A and B are respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 12.5 cm, PA = 5
cm, BR= 6 cm and PB = 4 cm. Is AB || QR? Give reasons for your answer.
4. In Fig, BD and CE intersect each other at the point P. Is PBC ~ PDE? Why?
5. In triangles PQR and MST, P = 55°, Q = 25°, M = 100° and S = 25°. Is QPR ~ TSM? Why?
6. Is the following statement true? Why?
“Two quadrilaterals are similar, if their corresponding angles are equal”.
7. Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the
perimeter of the other triangle. Are the two triangles similar? Why?
8. If in two right triangles, one of the acute angles of one triangle is equal to an acute angle of the other triangle,
can you say that the two triangles will be similar? Why?
3
9. The ratio of the corresponding altitudes of two similar triangles is . Is it correct to say that ratio of their
5
6
areas is 5
? Why?
10. D is a point on side QR of PQR such that PD QR. Will it be correct to say that PQD ~ RPD? Why?
12. Is it true to say that if in two triangles, an angle of one triangle is equal to an angle of another triangle and
two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar?
Give reasons for your answer.
1. In a Δ𝑃𝑄𝑅, 𝑃𝑅 2 – 𝑃𝑄 2 = 𝑄𝑅 2 and 𝑀 is a point on side 𝑃𝑅 such that 𝑄𝑀 ⊥ 𝑃𝑅. Prove that 𝑄𝑀2 = 𝑃𝑀 × 𝑀𝑅.
4. Diagonals of a trapezium 𝑃𝑄𝑅𝑆 intersect each other at the point 𝑂, 𝑃𝑄||𝑅𝑆 and 𝑃𝑄 = 3𝑅𝑆. Find the ratio of
the areas of triangles 𝑃𝑂𝑄 and 𝑅𝑂𝑆.
9. 𝐴𝐵𝐶𝐷 is a trapezium in which 𝐴𝐵||𝐷𝐶 and 𝑃 and 𝑄 are points on 𝐴𝐷 and 𝐵𝐶, respectively such that 𝑃𝑄||𝐷𝐶.
If 𝑃𝐷 = 18 cm, 𝐵𝑄 = 35 cm and 𝑄𝐶 = 15 cm, find 𝐴𝐷.
10. Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is
48 cm2 , find the area of the larger triangle.
11. In a triangle 𝑃𝑄𝑅, 𝑁 is a point on 𝑃𝑅 such that 𝑄𝑁 ⊥ 𝑃𝑅. If 𝑃𝑁. 𝑁𝑅 = 𝑄𝑁 2 , prove that ∠𝑃𝑄𝑅 = 90°.
12. Areas of two similar triangles are 36 cm2 and 100 cm2. If the length of a side of the larger triangle is 20 cm,
find the length of the corresponding side of the smaller triangle.
14. A 15 metres high tower casts a shadow 24 metres long at a certain time and at the same time, a telephone
pole casts a shadow 16 metres long. Find the height of the telephone pole.
15. Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height
of the point on the wall where the top of the ladder reaches.
1. In Fig., if ∠𝐴 = ∠𝐶, 𝐴𝐵 = 6 cm, 𝐵𝑃 = 15 cm, 𝐴𝑃 = 12 cm and 𝐶𝑃 = 4 cm, then find the lengths of 𝑃𝐷 and 𝐶𝐷.
2. It is given that Δ𝐴𝐵𝐶 ~ Δ𝐸𝐷𝐹 such that 𝐴𝐵 = 5 cm, 𝐴𝐶 = 7 cm, 𝐷 = 15 cm and 𝐷𝐸 = 12 cm. Find the lengths
of the remaining sides of the triangles.
3. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two
sides are divided in the same ratio.
4. In Fig., if 𝑃𝑄𝑅𝑆 is a parallelogram and 𝐴𝐵||𝑃𝑆, then prove that 𝑂𝐶||𝑆𝑅.
5. A 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If
the foot of the ladder is moved 1.6 m towards the wall, then find the distance by which the top of the ladder
would slide upwards on the wall.
6. For going to a city 𝐵 from city 𝐴, there is a route via city 𝐶 such that 𝐴𝐶 ⊥ 𝐶𝐵, 𝐴𝐶 = 2𝑥 km and 𝐶𝐵 = 2(𝑥 + 7)
km. It is proposed to construct a 26 km highway which directly connects the two cities 𝐴 and 𝐵. Find how
much distance will be saved in reaching city 𝐵 from city 𝐴 after the construction of the highway.
13. In fig., 𝑙 || 𝑚 and line segments 𝐴𝐵, 𝐶𝐷 and 𝐸𝐹 are concurrent at point P. Prove that
𝐴𝐸 𝐴𝐶 𝐶𝐸
= =
𝐵𝐹 𝐵𝐷 𝐹𝐷
14. In Fig., 𝑃𝐴, 𝑄𝐵, 𝑅𝐶 and 𝑆𝐷 are all perpendiculars to a line 𝑙, 𝐴𝐵 = 6 cm, 𝐵𝐶 = 9 cm, 𝐶𝐷 = 12 cm and 𝑆𝑃 = 36
cm. Find 𝑃𝑄, 𝑄𝑅 and 𝑅𝑆.
15. O is the point of intersection of the diagonals 𝐴𝐶 and 𝐵𝐷 of a trapezium 𝐴𝐵𝐶𝐷 with 𝐴𝐵||𝐷𝐶. Through 𝑂, a
line segment 𝑃𝑄 is drawn parallel to 𝐴𝐵 meeting 𝐴𝐷 in 𝑃 and 𝐵𝐶 in 𝑄. Prove that 𝑃𝑂 = 𝑄𝑂.
16. In Fig., line segment 𝐷𝐹 intersect the side 𝐴𝐶 of a triangle 𝐴𝐵𝐶 at the point
𝐵𝐷 𝐵𝐹
𝐸 such that 𝐸 is the mid-point of 𝐶𝐴 and ∠𝐴𝐸𝐹 = ∠𝐴𝐹𝐸. Prove that 𝐶𝐷 = 𝐶𝐸 .
17. Prove that the area of the semicircle drawn on the hypotenuse of a right angled triangle is equal to the sum
of the areas of the semicircles drawn on the other two sides of the triangle.
18. Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to
the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.