In a right-angled triangle with legs 6 cm and 10 cm, a rectangle of the largest area with a common angle

In a right-angled triangle with legs 6 cm and 10 cm, a rectangle of the largest area with a common angle is inscribed. Find the area of a rectangle.

Let’s denote by x and y the sides of the rectangle. The tangent of the angle of the triangle is: 6: 10 = 3/5. Then y = (10 – x) * 3/5. The area S is equal to:

S = x * y = 3/5 * x * (10 – x).

Let us examine the resulting function.

S ‘= (6x – 3 / 5x ^ 2)’ = 6 – 6/5 x.

6 – 6 / 5x = 0;

x = 5.

Then y = (10 – 5) * 3/5 = 3.