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In a right triangle, the two legs have lengths p and q and the hypotenuse has length h. What is the area of the triangle?
(1) h equals 10.
(2) p plus q equals 14.
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Correct answer
C
The area is (1/2)pq, so the whole task is to find the product pq. For a right triangle the legs also satisfy p² + q² = h².
(1) alone: h = 10 gives p² + q² = 100, one equation in two unknowns. Legs 6 and 8 give area 24; legs that satisfy p² + q² = 100 in other ways give other products. insufficient.
(2) alone: p + q = 14 is also one equation in two unknowns. Legs 7 and 7 give product 49; legs 4 and 10 give product 40. different products, different areas. insufficient.
together: from p + q = 14, squaring gives p² + 2pq + q² = 196. Substitute p² + q² = 100 to get 100 + 2pq = 196, so 2pq = 96 and pq = 48. The area is half of that, 24. We never needed to separate p from q. Answer: (C).
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