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505-555 (Easy)|   Geometry|      
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MathRevolution
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If ABCD is a square with its side length is 10ft and points P, Q, M ar 09 Sep 2016, 17:15
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89% (01:36) correct 11% (01:34) wrong based on 64 sessions

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If ABCD is a square with its side length is 10ft and points P, Q, M are midpoints of AD, AB, and BC, respectively, what is the area of region shaded as above figure?
A. 40
B. 45
C. 50
D. 75/2
E. 33
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D

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If ABCD is a square with its side length is 10ft and points P, Q, M ar 10 Sep 2016, 03:13
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MathRevolution
If ABCD is a square with its side length is 10ft and points P, Q, M are midpoints of AD, AB, and BC, respectively, what is the area of region shaded as above figure?
A. 40
B. 45
C. 50
D. 75/2
E. 33
*An answer will be posted in 2 days
Show more

We have information P,Q and M are midpoints of a square . Join midpoints P and M and then draw a perpendicular line from Q to X then we can split the square as per the below figure.

Now PDMC - a rectangle , then shaded region is 1/2*5*10 = 50/2

Now APQX is small square, then shaded region is 1/2*5*5 = 25/2.

Sum = 50/2+25/2 = 75/2.

Hence proved. Option D is the correct answer..
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If ABCD is a square with its side length is 10ft and points P, Q, M ar 10 Sep 2016, 13:16
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And what about square QXMB, which is 25 or 25/2.?
So the total square is 125/2.
Where is my mistake?
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If ABCD is a square with its side length is 10ft and points P, Q, M ar 10 Sep 2016, 13:48
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And what about square QXMB, which is 25 or 25/2.?
So the total square is 125/2.
Where is my mistake?
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Try this approach if you wish

Attachment:
1.png
1.png [ 30.61 KiB | Viewed 2901 times ]

Area of Δ AOP =\(\frac{1}{2}\) *\(5*5\) = \(\frac{25}{2}\)\(ft^{2}\)

Area of Δ MCD = \(\frac{1}{2}\) *\(5*10\) = \(50ft^{2}\)


Area of Δ AOP + Area of Δ MCD = \(\frac{25}{2}\)\(ft^{2}\) + \(\frac{1}{2}\) *\(5*10\) = \(50ft^{2}\) =>\(\frac{75}{2}\) \(ft^{2}\)

Hence answer will be (D)
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If ABCD is a square with its side length is 10ft and points P, Q, M ar 11 Sep 2016, 18:25
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The area of a triangle CDM is 25 and the area of a triangle APO is 25/2. The sum of areas is 25+(25/2)=75/2. Hence, the correct answer is D.
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If ABCD is a square with its side length is 10ft and points P, Q, M ar 06 Feb 2025, 17:09
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