Exemplar Exercise 6.3 (Triangles)

In a ∆ PQR, PR² – PQ² = QR² and M is a point on side PR such that QM ⊥ PR. Prove that

QM² = PM × MR.

Find the value of x for which DE || AB in Fig. 6.8

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In Fig. 6.9, if ∠1 = ∠2 and △NSQ ≅ △MTR, then prove that △PTS ~ △PRQ

Diagonals of a trapezium PQRS intersect each other at the point O, PQ and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS.

In Fig. 6.10, if AB || DC and AC and PQ intersect each other at the point O, prove that OA . CQ  = OC . AP.

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Find the altitude of an equilateral triangle of side 8 cm.

If ∆ABC ~ ∆DEF,  AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm, find the perimeter of ∆ABC.

In Fig. 6.11,  if DE || BC, find the ratio of ar (ADE) and ar (DECB).

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ABCD is a trapezium in which AB || DC and P and Q are points of AD and BC, respectively such that PQ || DC. If PD = 18 cm, BQ = 35 cm and QC = 15 cm, find AD.

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.

In a triangle PQR, N is a point on PR such that QN ⊥ PR. If PN × NR = QN², prove that ∠PQR = 90°.

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.

In Fig. 6.12, if ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, find BD.

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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.

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.