The hypotenuse of a right triangle is 6 1/2 centimetres and its area is 7 1/2 square centimetres. Calculate the lengths of its perpendicular

Question

The hypotenuse of a right triangle is 6\(\frac12\) centimetres and its area is 7\(\frac12\) square centimetres. Calculate the lengths of its perpendicular sides.

Answer 1

The perpendicular sides are x and y. Given that,

\(x^2+y^2=(6\frac{1}{2})^2\)

\(x^2+y^2=(\frac{13}{2})^2\)

\(x^2+y^2=\frac{169}{4}\)----(1)

\(\frac{1}{2}x\times y=7\frac{1}{2};\)

xy = 15 ----(2)

From (1) and (2)

\((x+y)^2=x^2+y^2+2xy\)

\((x+y)^2=\frac{169}{4}+2\times15=\frac{169}{4}+30=\frac{289}{4}\)

x + y = \(\frac{17}{2}\) -----(3)

(x - y)² = x² + y² - 2xy

\(=\frac{169}{4}-30=\frac{49}{4}\)

From (3) and (4)

2x = 24/2 = 12

∴ x = 6

6 – y = 7/2

∴ y = 5/2 = 2.5

∴ Perpendicular sides = 6 and 2.5