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Triangles

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371. In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, Prove that the angle opposite the first side is a right angle. Use the above theorem and prove the following.
In triangle ABC, AD ⊥ BC and BD = 3 CD. Prove that 2 AB² = 2 AC² + BC².

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372.

State and Prove Converse of Pythagoras Theorem.
Use the above theorem to prove the following:
In a quadrilateral ABCD, ∠B = 90. If AD²= AB² + BC² + CD², prove that ∠ACD = 90°.

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373. State and Prove Basic Proportionality theorem. Using the above, do the following in Fig. PQ || AB and PR || AC showthat QR = BC.

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374. Prove that the ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Use the above theorem on the following: ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm, prove that area of ∆APQ is one-sixteenth of the area of ∆ABC.

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375.

The ratio of the areas of similar triangles is equal to the ratio of the squares on the corresponding sides. Prove it.
Using the above theorem, prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonals.

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376.

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the square on the other two sides. Prove
Using the above theorem, determine the length of AD in terms of b and c.

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377.

In a right triangle, prove that the square on the hypotenuse is equal to sum of the squares on the other two sides.
Using the above result, prove the following: In the Fig, PQR is a right triangle, right angled at Q. If QS = SR, show that PR² = 4PS² - 3PQ².