In trapezoid PQRS, segment P Q is parallel to segment S R. Find the area of PQRS. Leave your answer in simplest radical form.
The image shows trapezoid P Q R S.P Q equals 18.
Q R equals 24.
angle S equals 135 degrees.
angle R equals 30 degrees.
A dashed segment connects point Q at a right angle with S R.
The distance on the bottom side from S to the right dashed segment is 6.
(1 point)
Responses
144 + 72Start Root 3 End Root
144 + 72 Image with alt text: Start Root 3 End Root
72 + 72Start Root 3 End Root
72 + 72 Image with alt text: Start Root 3 End Root
288Start Root 3 End Root – 216
288 Image with alt text: Start Root 3 End Root – 216
144Start Root 3 End Root – 72
To find the area of trapezoid PQRS, we can split it into two triangles, triangle PQS and triangle SQR.
First, let's find the height of the trapezoid. Since QP is parallel to RS, we can extend the dashed segment to meet segment PS at point T. This forms a right triangle PTS.
Since angle S is 135 degrees and angle R is 30 degrees, angle T is 180 - 135 - 30 = 15 degrees. This means that triangle PTS is a 30-60-90 triangle.
Let's label the length of PT as h (the height of the trapezoid). Since PS is the longer leg in a 30-60-90 triangle, PS = 2h.
Now let's focus on triangle PQS. We know that PS = 2h and PQ = 18. To find the area of triangle PQS, we use the formula A = (1/2) * base * height, where the base is PQ and the height is PS.
A(PQS) = (1/2) * PQ * PS
= (1/2) * 18 * 2h
= 18h
Next, let's focus on triangle SQR. We know that SR = 24 and QR = 18. To find the area of triangle SQR, we again use the formula A = (1/2) * base * height, where the base is QR and the height is SR.
A(SQR) = (1/2) * QR * SR
= (1/2) * 18 * 24
= 216
To find the area of the trapezoid PQRS, we add the areas of triangles PQS and SQR.
A(PQRS) = A(PQS) + A(SQR)
= 18h + 216
Now, we need to find the value of h. Since triangle PTS is a 30-60-90 triangle, we can use the ratios of the sides to find h.
In a 30-60-90 triangle, the ratio of the lengths of the sides is 1 : √3 : 2.
Since PS is the longer leg (the side opposite the 60-degree angle), PS/PT = √3/1.
Thus, 2h/h = √3/1.
Simplifying, we get 2 = √3.
Squaring both sides, we get 4 = 3.
Subtracting 3 from both sides, we get 1 = 0.
This is not a valid equation, so there is no real value for h. However, we can still find the area of the trapezoid in terms of h.
A(PQRS) = 18h + 216
So, the area of trapezoid PQRS is 18h + 216, where h is the height of the trapezoid.