Question

Isocles trapazoid abcd is inscribed in a circle , the longer base is two times as long as the shorter base and the diagonal is the angle bisector for the acute angle. find the measures of all angles in the trapazoid, the lengths of the bases, and the radius of the circle if the length of leg is 24ft

Answer 1

The measures of all angles in the trapezoid are: Angle A: 90 degrees (right angle); Angle B: 90 degrees (right angle); Angle C: 45 degrees; Angle D: 45 degrees. The shorter base of the trapezoid is approximately 13.86ft and the longer base is approximately 27.72ft. The radius of the circle is approximately 10.89ft.

Step-by-step explanation:

The measures of all angles in the trapezoid are:

• Angle A: 90 degrees (right angle)
• Angle B: 90 degrees (right angle)
• Angle C: 45 degrees
• Angle D: 45 degrees

The shorter base of the trapezoid is denoted as a, and the longer base is 2a. The diagonal acts as the angle bisector, so it divides angle C into two equal angles, each measuring 22.5 degrees. The length of the leg is given as 24ft.

Using the Pythagorean theorem, we can find the length of the shorter base, a:

a² + 24² = (2a)²

a² + 576 = 4a²

3a² = 576

a² = 192

a ≈ 13.86

Therefore, the shorter base (base AD) is approximately 13.86ft and the longer base (base BC) is approximately 27.72ft.

The radius of the circle can be found using the formula:

r = (AD + BC) / 4

r = (13.86 + 27.72) / 4

r = 10.89

Therefore, the radius of the circle is approximately 10.89ft.