Question

Isosceles trapezoid 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 of the trapezoid, the lengths of the bases, and the radius of the circle if the length of the leg is 24 ft.

a) m/a=m/b=m/c=m/d=ab=ft
b) bc=ad=ft
c) cd=ft
d) r=ft

Answer 1

To find the measures of the angles in the isosceles trapezoid, we can use the fact that the diagonal is the angle bisector of the acute angle. Opposite angles in inscribed trapezoids are supplementary. From there, we can determine the measures of all angles in the trapezoid.

Step-by-step explanation:

Let's denote the length of the shorter base as a. With the longer base being two times as long as the shorter base, the length of the longer base can be denoted as 2a. Since the diagonal is the angle bisector of the acute angle, we can consider triangle ADC. In this triangle, angle CAD is bisected by diagonal AC. From this, we can conclude that angle ACD is half the measure of the acute angle.

Since trapezoid ABCD is inscribed in a circle, opposite angles are supplementary. Therefore, angle BAD is equal to angle ACD. This allows us to determine the measures of all angles of the trapezoid.

• Angle ACD = angle CAD = half the measure of the acute angle
• Angle CAB = angle BCD = supplementary to angle CAD
• Angle ABD = angle DCB = supplementary to angle ACD

Using the information above, we can determine the measures of all the angles in the trapezoid.