In the diagram, the area of rectangle $P Q R S$ is 24 . If $T Q=T R$, the area of quadrilateral $P T R S$ is
(A) 18
(B) 20
(C) 16
(D) 6
(E) 15

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-077.jpg?height=239&width=420&top_left_y=1710&top_left_x=1292)
Solution 1

Since the area of rectangle $P Q R S$ is 24 , let us assume that $P Q=6$ and $Q R=4$.

Since $Q T=Q R$, then $Q R=2 Q T$ so $Q T=2$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-336.jpg?height=266&width=431&top_left_y=360&top_left_x=912)

Therefore, triangle $P Q T$ has base $P Q$ of length 6 and height $Q T$ of length 2, so has area $\frac{1}{2}(6)(2)=\frac{1}{2}(12)=6$.

So the area of quadrilateral $P T R S$ is equal to the area of rectangle $P Q R S$ (which is 24 ) minus the area of triangle $P Q T$ (which is 6 ), or 18 .

## Solution 2

Draw a line through $T$ parallel to $P Q$ across the rectangle parallel so that it cuts $P S$ at point V.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-336.jpg?height=261&width=441&top_left_y=1108&top_left_x=904)

Since $T$ is halfway between $Q$ and $R$, then $V$ is halfway between $P$ and $S$.

Therefore, $S V T R$ is a rectangle which has area equal to half the area of rectangle $P Q R S$, or 12.

Similarly, $V P Q T$ is a rectangle of area 12 , and $V P Q T$ is cut in half by $P T$, so triangle $P V T$ has area 6 .

Therefore, the area of $P T R S$ is equal to the sum of the area of rectangle $S V T R$ and the area of triangle $P V T$, or $12+6=18$.

ANSWER: (A)