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5. In trapezoid PQRS, . Find the area of PQRS. Leave your answer in simplest radical form. (1 point) 144 + 72sqrt3 72 + 72sqrt3 288sqrt3 – 216 144sqrt3 – 72
With triangle RTQ, we see that angle T is 90° and angle R is 30°. Since the angles in a triangle always sum to 180°, angle Q is 180° - 90° - 30° = 60°. Thus, triangle RTQ is a 30°-60°-90° right triangle. The sides of a 30°-60°-90° triangle are in ratio 1:√3:2 (this is in the form (side opposite to 30° angle):(side opposite to 60° angle):(side opposite to 90° angle)). The hypotenuse, the side opposite opposite to the 90° angle, has a length of 24. The altitude of the triangle, the side opposite to the 30° angle, therefore has a length of 24/2 = 12; and the base of the triangle, the side opposite to the 60° angle, has a length of 12√3. Therefore, the area of PQRS is: (area of left trapezoid) + (area of triangle) = (1/2)(6 + 18)(12) + (1/2)(12√3)(12) = 144 + 72√3.