2016 · Q51
Suppose chords AB and CD of a circle intersect at a point P inside the circle. Two right-angled triangles A′P′B′ and C′Q′D′ are formed as shown in the figures below such that A′P′ = AP, B′P′ = BP, C′Q′ = CP, D′Q′ = DP and ∠A′P′B′ = 90° = ∠C′Q′D′. Which of the following statements are not correct?
- 1A′P′B′ and C′Q′D′ are similar triangles, but need not be congruent.
- 2A′P′B′ and C′Q′D′ are congruent triangles.
- 3A′P′B′ and C′Q′D′ are triangles of same area.
- 4A′P′B′ and C′Q′D′ are triangles of same perimeter.
Elementary MathematicsCircles
- A2 and 3 only
- B1 and 3 only
- C1, 2 and 4 only
- D1, 2, 3 and 4
Explanation
By the intersecting-chords theorem AP·BP = CP·DP, so the right triangles have equal area ½·AP·BP = ½·CP·DP — statement 3 is true. But AP/CP = DP/BP, not BP/DP, so ΔA'P'B' is not similar to ΔC'Q'D' in the stated correspondence (statement 1 false), hence not congruent (statement 2 false) and perimeters differ (statement 4 false). The incorrect statements are 1, 2 and 4, option C.