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# In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area

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Math Expert
Joined: 02 Sep 2009
Posts: 44287
In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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11 Oct 2017, 00:46
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Question Stats:

75% (01:32) correct 25% (00:44) wrong based on 53 sessions

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In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area of the quadrilateral PRUT?

(A) 1/8
(B) 1/4
(C) 3/8
(D) 5/8
(E) 2/3

[Reveal] Spoiler:
Attachment:

2017-10-11_1127.png [ 4.3 KiB | Viewed 845 times ]
[Reveal] Spoiler: OA

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Re: In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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11 Oct 2017, 01:10
Area of PQRS = 1 ==> Areas of PRS=1/2
We know that iaw similarity properties:
Area of PRS/Area of TUS = RS^2/US^2= 1^2/0,5^2=1/0,25
Area of TUS = 1/8, therefore Area of PRTU = Area of PRS - Area of TUS = 1/2-1/8=3/8

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Re: In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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11 Oct 2017, 01:14
Side QR=1
Area of square PQRS = 1^2 =1
Area of triangle PRS = 1/2 of PQRS = 1/2
Area of triangle UTS = 1/2 * ST * SU = 1/2 *1/2 * 1/2 = 1/8
Area of quadrilateral PRUT = Area of triangle PRS - Area of triangle UTS =1/2 - 1/8 = 3/8
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Re: In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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13 Oct 2017, 09:29
Bunuel wrote:

In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area of the quadrilateral PRUT?

(A) 1/8
(B) 1/4
(C) 3/8
(D) 5/8
(E) 2/3

[Reveal] Spoiler:
Attachment:
2017-10-11_1127.png

The area of square QRPS is 1 x 1 = 1.

The area of triangle QRP is 1/2.

The area of triangle TUS is 1/2 x 1/2 x 1/2 = 1/8.

Thus, the area of the quadrilateral PRUT is 1 - (1/2 + 1/8) = 1 - (4/8 + 1/8) = 3/8.

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Re: In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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15 Oct 2017, 05:42
Area of PRUT = Area of PQRS - Area of UST - Area of PQR

Area of PQRS = 1*1 = 1

Since QR = PQ = 1 so Area of PQR = 1/2 * 1 * 1 = 1/2

Since RU = US = ST = PT = 1/2 so Area of UST = 1/2 * 1/2 * 1/2 = 1/8

So the area of PRUT = 1 -1/2 -1/8 = 3/8

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Re: In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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15 Oct 2017, 06:40
Bunuel wrote:

In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area of the quadrilateral PRUT?

(A) 1/8
(B) 1/4
(C) 3/8
(D) 5/8
(E) 2/3

[Reveal] Spoiler:
Attachment:
2017-10-11_1127.png

If we divide square into triangles it would be 8 triangles of equal area
in shaded portion there are 3 triangles so ea of Quad is 3/8
VP
Joined: 22 May 2016
Posts: 1416
In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area [#permalink]

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15 Oct 2017, 08:50
Bunuel wrote:

In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area of the quadrilateral PRUT?

(A) 1/8
(B) 1/4
(C) 3/8
(D) 5/8
(E) 2/3

[Reveal] Spoiler:
Attachment:
2017-10-11_1127.png

$$(Area_{square}) - (Area_{2triangles}) = (Area_{quad})$$

Square side length = 1
Top triangle legs' length = 1
Bottom triangle legs' length* = $$\frac{1}{2}$$

Area of square: $$s^2 = 1^2 = 1$$
Area of top triangle: $$\frac{s^2}{2}=\frac{1}{2}$$
Area of bottom triangle: $$\frac{s^2}{2} =\frac{(\frac{1}{4})}{2} = \frac{1}{8}$$

Both triangles' area:
$$\frac{1}{2} + \frac{1}{8} = \frac{5}{8}$$

Quadrilateral area: $$1 - \frac{5}{8}=\frac{3}{8}$$

*The segments that constitute the legs of this right triangle are both $$\frac{1}{2}$$ the length of the side of the square because they are half of a bisected side.
Square side RS is bisected; we are given that RU = US. Square side PS is similarly bisected.

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In square PQRS above, QR = 1, RU = US, and PT = TS. What is the area   [#permalink] 15 Oct 2017, 08:50
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