An isosceles trapezoid is described around a circle with a diameter of 15 cm and the side side is 17 cm.

An isosceles trapezoid is described around a circle with a diameter of 15 cm and the side side is 17 cm. Find the base of the trapezoid.

The height of a trapezoid circumscribed about a circle is equal to the diameter of this circle:

h = 15 cm.

Since this isosceles trapezoid is circumscribed about a circle, the sums of the lengths of the opposite sides of the trapezoid are equal. This means that the sum of the lengths of equal lateral sides is equal to the sum of the lengths of the bases a and b:

a + b = 17 * 2 = 34 cm.

Consider a right-angled triangle formed by the height of the trapezoid h, the lateral side c, and the projection of the lateral side onto the larger base. By the Pythagorean theorem, the square of the projection is:

c2 – h2 = 172 – 152 = 289 – 225 = 64 = 82.

The side projection is 8 cm.

The length of the larger base b is equal to the sum of the lengths of the smaller base a and two projections of the lateral sides, each of which is 8 cm:

b = a + 8 + 8 = a + 16.

Substituting this expression for b into the equation a + b = 34, we get:

a + a + 16 = 34;

2 * a = 18;

a = 18/2 = 9 cm – smaller base.

b = a + 16 = 9 + 16 = 25 cm – larger base.