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Sub 505 (Easy)|   Geometry|      
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Bunuel
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 00:15
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E
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In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128


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E
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 01:24
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Bunuel

In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128
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There are a couple of ways to solve this.

1) Since the square QRST is formed by joining the mid points of square ABCD, then Area of QRST = 1/2 the area of ABCD

Area of ABCD = 16 * 16 = 256. Therefore area of QRST = 256/2 = 128


2) Triangle ARQ is an isosceles right angled triangle, with AQ = AR = 16/2 = 8

using the Pythagoras Theorem, \(QR^2 = 8^2 + 8^2 = 2*8^2\). So QR = \(8\sqrt{2}\) and this is the side of the smaller square.

Area = \(8\sqrt{2} \space * \space 8\sqrt{2} = 128\)


Option E

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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 01:58
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Bunuel

In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128


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Here's how i solved it.

Since ABCD is a square = All the 4 angles = 90 degrees.

Side of the square = 16

where R, S ,T ,Q are mid points which AQ = QD and same for others.

Using pythagorus theorem because Traingle QAR Is right angle triangle

AQ^2 * AR^2 = QR^2
8^2 + 8 ^2 = QR^2
64 + 64 = QR^2
8√2 = QR

One side = QR = 8√2

Area of Square = a^2 = (8√2)^2 = 128

Answer E
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 02:17
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In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128
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Diagonal is 16

So, Area is \(\frac{16*16}{2}=128\) {AS AREA OF A SQUARE IS 1/2 * DIAGONAL\(^2\)}

Thus, Answer must be (E) 128
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 07:45
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Since ABCD has an area of 256
and R, S, T, and Q are midpoints of the sides of ABCD.

hence RSTQ will also form a square exactly half of the area of ABCD since half the portion is left out in ABCD when forming RSTQ.

hence IMO area of RSTQ will be exactly half of the area of ABCD.

hence the area of RSTQ will be 256/2 = 128
Hence Option E


Bunuel

In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128


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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 08:27
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Given:
ABCD is a square with side length 16
R, S, T, and Q are midpoints of the sides of ABCD

Need:
What is the area of RSTQ?

Length of AB is 16 and R is midpoint of side AB, so AR=RB=8
Similarly BS=SC=8

The formula to calculate length of 3rd side RS is \(RB^2\) + \(BS^2\) = \(RS^2\)
\(8^2\) + \(8^2\) = \(RS^2\)
64+64 = \(RS^2\)
128=\(RS^2\)
\(8\sqrt{2}\) = RS

We got side length of one side of the square RSTQ, based on which we can calculate the area of square RSTQ

Area of RSTQ = \(S^2\) = \((8\sqrt{2})^2\) = 128

Area of square RSTQ = 128

Ans: E
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 11:18
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Bunuel

In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128

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each side of RSTQ = (8^2 + 8^2)^1/2 =8(2)^1/2

area of RSTQ = (8(2)^1/2)^2 = 128

Answer E
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 30 Oct 2020, 21:45
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In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

ABCD is a square with side length 16.

R, S, T, and Q are midpoints of the sides of ABCD.

When midpoints of the square ABCD are joined, a square QRST is formed with diagonal RT = AD = 16

Area of QRST = (1/2)*d^2
= (1/2)*16*16
= 128

Choice E
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In the figure shown, ABCD is a square with side length 16. R, S, T, an 31 Oct 2020, 12:07
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Bunuel

In the figure shown, ABCD is a square with side length 16. R, S, T, and Q are midpoints of the sides of ABCD. What is the area of RSTQ?

A. 64
B. 72
C. 112
D. 121
E. 128


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Given: ABCD is a square with R, S, T, and Q as midpoints.
=> DQT will be a right angled triangle with base and height as 8 and 8 respectively.
=> since R,S,T,Q are all mid-points, triangles AQR, RBS, SCT, DQT will have the same base and height => all 4 triangles will have the same area.

Area of triangle DQT= 1/2* (8*8) = 32.
=> Sum of area of all 4 triangles is = 32*4 = 128.
Area of the square ABCD =256.
Remaining area (Area of RSTQ) = Area of ABCD - Area of the 4 equal right angled triangles => 256-128 = 128.

Answer is E.
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