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Question:

The altitude of a right triangle is 7cm less than its base. If the hypotenuse is 13cm, find the other two sides.

Solution:

Let the base be x cm. Then the altitude will be x-7 cm.
By Pythagoras theorem, x² + (x-7)² = 13²
→ x² + x² - 14x + 49 = 169
→ 2x² - 14x - 120 = 0
→ x² - 7x - 60 = 0
→ x² - 12x + 5x - 60 = 0
→ x(x - 12) + 5(x - 12) = 0
→ (x + 5)(x - 12) = 0
→ x = 12, -5
Length cannot be negative, then the base is 12 cm and the height is 5 cm