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21 Dec 2021, 07:07
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
74% (02:29) correct
26%
(02:54)
wrong
based on 90
sessions
History
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ABCD is a Square and OEH and OFG are equilateral triangles, where O is the point of intersection of two lines GH and EF. If EH = GF = 1 cm, what is the sum of area of OHBCF and OGDAE?
A.\( 3-\frac{\sqrt{3}}{4} \)
B. \(3-\frac{\sqrt{3}}{2} \)
C. \(3-\sqrt{2} \)
D. \(\frac{\sqrt{3}}{2} \)
E. \(\frac{\sqrt{3}}{4}\)
Attachment:
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21 Dec 2021, 09:19
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ktzsikka
Side of Square ABCD = 2*Height of equilateral triangle with side 1 \(= 2*(\frac{√3}{2})1 = √3\)
Area of ∆OGF = Area of ∆ OEH \(= (\frac{√3}{4})*1^2 = (\frac{√3}{4})\)
sum of area of OHBCF and OGDAE = Area of square - Area of ∆ OEH - Area of ∆OGF \(= (√3)^2 - 2*(\frac{√3}{4}) = 3-(\frac{√3}{2})\)
Answer: Option B
28 Dec 2021, 10:55
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Here is the OE
Solution:
Step 1: Understand Question Statement
• ABCD is a square.
• OEH and OFG are equilateral triangles.
• EH = GF = 1 cm
We need to find Area of OHBCF + Area of OGDAE.
Step 2: Define Methodology
• Since OEH and OFG are equilateral triangles and EH = GF = 1 cm.
• We can easily find the area of OEH and GOF.
• Now, if we can find the area of the square, we can write:
• Area of OHBCF + Area of OGDAE = Area of Square – Area of OEH – Area of GOF.
So, let’s try to find area of square.
Step 3: Calculate the final answer
Notice that if we drop a perpendicular from O to both the triangles, we easy visualize that XY = AD = BC = side of square.
• In triangle OEH, the height OY =\(\frac{\sqrt{3}}{2}*a\) , here EH = a = 1
• XY = 2* OY, since both equilateral triangles are equal.
• AD = XY = 2* OY = \(2* \frac{\sqrt{3}}{2}*a\)=\(\sqrt{3}*1= \sqrt{3}\)
Thus, Area of OHBCF + Area of OGDAE = Area of Square – Area of OEH – Area of GOF.
Area of OHBCF + Area of OGDAE = \((\sqrt{3})^2\) - \(\frac{\sqrt{3}}{4}*1^2\) - \(\frac{\sqrt{3}}{4}*1^2\) = \(3-\frac{\sqrt{3}}{2}\)
Hence, the correct answer is Option B.
Solution:
Step 1: Understand Question Statement
• ABCD is a square.
• OEH and OFG are equilateral triangles.
• EH = GF = 1 cm
We need to find Area of OHBCF + Area of OGDAE.
Step 2: Define Methodology
• Since OEH and OFG are equilateral triangles and EH = GF = 1 cm.
• We can easily find the area of OEH and GOF.
• Now, if we can find the area of the square, we can write:
• Area of OHBCF + Area of OGDAE = Area of Square – Area of OEH – Area of GOF.
So, let’s try to find area of square.
Step 3: Calculate the final answer
Notice that if we drop a perpendicular from O to both the triangles, we easy visualize that XY = AD = BC = side of square.
• In triangle OEH, the height OY =\(\frac{\sqrt{3}}{2}*a\) , here EH = a = 1
• XY = 2* OY, since both equilateral triangles are equal.
• AD = XY = 2* OY = \(2* \frac{\sqrt{3}}{2}*a\)=\(\sqrt{3}*1= \sqrt{3}\)
Thus, Area of OHBCF + Area of OGDAE = Area of Square – Area of OEH – Area of GOF.
Area of OHBCF + Area of OGDAE = \((\sqrt{3})^2\) - \(\frac{\sqrt{3}}{4}*1^2\) - \(\frac{\sqrt{3}}{4}*1^2\) = \(3-\frac{\sqrt{3}}{2}\)
Hence, the correct answer is Option B.
Attachments
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