Question

The sides (other than hypotenuse) of a right triangle are in the ration 3:4. A rectangle is described on its hypotenuse, the hypotenuse being the longer side of the rectangle. The breadth of the rectangle is 4/5 of its length. Find the shortest side of the right triangle if the perimeter of the rectangle is 180cm.

Answer 1

right triangle is a 3:4:5 triangle since this is a special triangle.
without knowing that, the calculation of a^2 + b^2 = c^2 would yield the same conclusion, i.e. that the ratio is 3:4:5.
since the hypotenuse of the triangle and the rectangle length are the same, then the rectangle length is assigned 5 to start with.
since the rectangle width is 4/5 the rectangle length, then the rectangle width is assigned a 4 initially.
perimeter of rectangle is 180 cm.
let x = width of rectangle and 5/4 * x = length of rectangle since they are in this ratio.
perimeter = 2 * length + 2 * width, so p = (2 * (5/4) * w) + (2 * w)
this winds up being p = 18 * x / 4 so x = 4 * p / 18.
since p = 180, then x = 4 * 180 / 18 = 4 * 10 = 40.
5/4 x = 50 which is also the length of the hypotenuse.
that makes the other sides of the triangle 40 and 30 because of their ratio to the hypotenuse (4/5 and 3/5 respectively).
so the answer to the question is 30 which is the shortest side of the triangle.