In the figure above, what is the area of region PQRST ?

[AbeNeedsAnswers]

In the figure above, what is the area of region PQRST ?

(1) PQ = RS
(2) PT = QT

C

Source: Official Guide 2020

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Hi All,

We're asked for the area of region PQRST. This question is based around a couple of Geometry rules. To properly find the area, we can break this shape into 2 pieces (a rectangle and a triangle), so we need to know the dimensions of the rectangle and the exact type of triangle - since we have one of the sides, we either need the 3 angles or the 2 missing sides - to calculate the overall area.

(1) PQ = RS

With the information in Fact 1, we know that the width of the rectangle is 6 and one of the two missing triangle sides is also 6. Along with the diagonal of the rectangle, we can now calculate its length (it's 8, since we have a 6/8/10 right triangle in the rectangle) but without the 3rd side of the triangle (or its 3 angles), we cannot calculate that other area.
Fact 1 is INSUFFICIENT

(2) PT = QT

The information in Fact 2 tells us that the triangle is either Isosceles or Equilateral, but we still do not know enough to determine its area and we don't know the length or width of the rectangle, so we cannot determine its area either.
Fact 2 is INSUFFICIENT

Combined, we know...
PQ = RS
PT = QT

With both Facts, we know that the triangle's sides are 6/6/6, so it's Equilateral and we can calculate its area. We also know the area of the rectangle (it's 6x8 = 48), so we CAN determine the area of PQRST.
Combined, SUFFICIENT

Final Answer: C

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Rich

[Scott@TargetTestPrep]

In the figure above, what is the area of region PQRST ?

(1) PQ = RS
(2) PT = QT

C

Source: Official Guide 2020

Solution:

Question Stem Analysis:

We need to determine the area of region PQRST. Notice that the area of the region is composed of triangle PQT and rectangle QRST.

Statement One Alone:

From statement one, we see that RS = 6. Since triangle RST is a right triangle with side RS = 6 and hypotenuse RT = 10, side ST = 8 (notice triangle RST is a 6-8-10 right triangle). Since RS and ST are also the sides of rectangle QRST, the area of rectangle QRST is 6 x 8 = 48. However, we can’t determine the area of triangle PQT, so we can’t determine the area of region PQRST. Statement one alone is not sufficient.

Statement Two Alone:

From statement two, we see that triangle PQT is at least an isosceles triangle and perhaps an equilateral triangle. However, since we don’t know which one it really is, we can’t determine its area. Statement two alone is not sufficient.

Statements One and Two Together:

With the two statements, we see that RS = QT and since RS = 6, QT = 6. Furthermore, PT = QT = PQ = 6. This makes triangle PQT an equilateral triangle. Since we know a side of the equilateral triangle, we can determine its area. Lastly, since we’ve already determined the area of rectangle QRST to be 48, we can determine the area of region PQRST. Both statements together are sufficient.

Answer: C