Exercise 6.3: Criteria for Similarity of Triangles

A student is asked to determine if a triangle with sides 7 cm, 24 cm, and 25 cm is a right-angled triangle. The student's solution: 'Let the sides be a=7, b=24, c=25. To check if it's a right-angled triangle, I'll use Pythagoras theorem: a² + c² = b². So, 7² + 25² = 24². 49 + 625 = 576. 674 = 576. Since 674 ≠ 576, it is not a right-angled triangle.' What is the mistake in the student's reasoning or calculation?

Two triangles ΔXYZ and ΔMNO have sides such that XY = 4 cm, YZ = 6 cm, ZX = 8 cm and MN = 6 cm, NO = 9 cm, OM = 12 cm. Are these triangles similar?

In ΔABC, D and E are points on AB and AC respectively such that DE || BC. If AD:DB = 2:3 and area of ΔADE = 16 cm², then what is the area of trapezium DBCE?

A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

In ΔABC, D and E are points on AB and AC respectively such that AD = 3 cm, DB = 5 cm, AE = 4.5 cm, and EC = 7.5 cm. If ∠ADE = 50°, and ∠ABC = 70°, what is the measure of ∠BAC?

Consider two triangles, ΔABC and ΔDEF. Which of the following statements is **incorrect**?

In ΔABC, D and E are points on AB and AC respectively. If AD = 3 cm, DB = 5 cm, AE = 4.5 cm, and EC = 7.5 cm, and ∠A = 60°, what is the measure of ∠AED?

A ladder 15 m long reaches a window 9 m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 12 m high. What is the width of the street?

In ΔABC, P and Q are points on AB and AC respectively. If AP = x, PB = x-2, AQ = x+2, QC = x-1, for what value of x will PQ be parallel to BC?

A ladder 15 m long is placed against a wall such that its foot is 9 m away from the wall. If the top of the ladder slips down by 3 m, how much further will its foot slide away from the wall?