The hypotenuse of a right-angled triangle is 13 cm. The height drawn to the hypotenuse is 6 cm.

The hypotenuse of a right-angled triangle is 13 cm. The height drawn to the hypotenuse is 6 cm. Find the legs of the triangle using the system of equations.

It is known:

Right triangle;
Hypotenuse = 13 cm;
Height drawn to the hypotenuse = 6 cm.
Let’s find the legs of a right-angled triangle through the system of equations.

Let x be the first leg of the triangle.

y – the second leg of the triangle.

13 = x1 + (13 – x1);

We get:

{6 ^ 2 = x1 * (13 – x1);

x ^ 2 = 13 * x1;

y ^ 2 = 13 * (13 – x);

{36 = x1 * (13 – x1);

x ^ 2 = 13 * x1;

y ^ 2 = 13 * (13 – x1);

{0 = -36 + x1 * 13 – (x1) ^ 2;

x ^ 2 = 13 * x1;

y ^ 2 = 13 * (13 – x1);

{x1 = 9 and x1 = 4;

The hypotenuse is 9 + 4 = 13.

x ^ 2 = 13 * 9 = 13 * 9;

x = 3√13;

y ^ 2 = 13 * (13 – 9) = 13 * 4;

y = 2√13;

This means that the legs of the rectangle are 3√13 and 2√13.