The diagonal of a rectangle inscribed in a circle is 10 cm, and its diagonal of a rectangle inscribed in a circle is 10 cm

The diagonal of a rectangle inscribed in a circle is 10 cm, and its diagonal of a rectangle inscribed in a circle is 10 cm, and its area is 48 cm, find the radius and sides of the rectangle.

It is known that the diagonal of a rectangle inscribed in a circle is equal to its diameter. Therefore, the radius of the circle is half the diagonal:

r = d / 2 = 10/2 = 5 cm.

Two adjacent sides of a rectangle and its diagonal form a right-angled triangle, for which, according to the Pythagorean theorem, we can write:

a ^ 2 + b ^ 2 = d ^ 2;

a ^ 2 + b ^ 2 = 100.

The area of ​​a rectangle is equal to the product of its adjacent sides:

a * b = 48.

Thus, we have a system of equations:

1) a ^ 2 + b ^ 2 = 100;

2) a * b = 48.

Multiplying both sides of the second equation by 2 and adding the result with the first equation, we get:

a ^ 2 + b ^ 2 + 2 * a * b = 100 + 48 * 2;

(a + b) ^ 2 = 196;

a + b = 14;

a = 14 – b.

Substituting the resulting expression for a into the second equation, we get:

(14 – b) * b = 48;

14b – b ^ 2 = 48;

b ^ 2 – 14b + 48 = 0.

Let’s solve the quadratic equation:

D = 14 ^ 2 – 4 * 1 * 48 = 196 – 192 = 4 = 22;

b1 = (14 – 2) / 2 = 6;

b2 = (14 + 2) / 2 = 8.

Thus, the sides of this rectangle are 6 cm and 8 cm.