GMAT Official Guide 2019 The figure above represents a
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A
B
C
D
E
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- Brent@GMATPrepNow
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Let's add the radius of 6 feet to the diagram to get:
From here, we can see the right triangle hiding within the diagram, which means we can apply the Pythagorean Theorem.
Se can write: h² + 2² = 6²
Simplify: h² + 4 = 36
So, we get: h² = 32
This means h = √32
Check the answer choices. . . √32 is not among them.
Looks like we need to simplify √32
We'll use the fact that √(xy) = (√x)(√y)
So, √32 = √[(16)(2)] = (√16)(√2) = (4)(√2) = 4√2
Check the answer choices. . . D
Answer: D
Cheers,
Brent
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- ceilidh.erickson
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Brent provided a nice solution, but there's also an easy way to GUESS on this problem:
Since h is not the radius but is perpendicular to the diameter, we can infer that it's shorter than the radius:
Therefore, we can eliminate E.
We can also infer that h must be greater than 4, since the angle that's across from must be greater than 45 degrees. Compare these two:
We don't have to calculate the angle of x to see that it's opening wider than 45*. Thus angle y must be less than 45 degrees. Since larger sides are across from larger angles, h must be larger than 4.
Eliminate any answer choices less than 4: A, B, and C.
The only thing left must be the answer: D.
Since h is not the radius but is perpendicular to the diameter, we can infer that it's shorter than the radius:
Therefore, we can eliminate E.
We can also infer that h must be greater than 4, since the angle that's across from must be greater than 45 degrees. Compare these two:
We don't have to calculate the angle of x to see that it's opening wider than 45*. Thus angle y must be less than 45 degrees. Since larger sides are across from larger angles, h must be larger than 4.
Eliminate any answer choices less than 4: A, B, and C.
The only thing left must be the answer: D.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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For another exact approach, you could use SIMILAR TRIANGLES:
We can create 2 similar right triangles inside the semicircle using h as a side length for each:
Since side lengths in similar triangles are proportional, we can infer:
(4/h) = (h/8)
h² = 32
h = √32
h = 4√2
We can create 2 similar right triangles inside the semicircle using h as a side length for each:
Since side lengths in similar triangles are proportional, we can infer:
(4/h) = (h/8)
h² = 32
h = √32
h = 4√2
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
Another way to solve without doing any math:
-You know h < 6; the height above the center of the circle is the highest point
-Draw a circle with radius 4 from the bottom of line h. It is a smaller arc and won't reach the top of the arc where "h" is, so you know h > 4
-Only option for 4 < h < 6 is D
-You know h < 6; the height above the center of the circle is the highest point
-Draw a circle with radius 4 from the bottom of line h. It is a smaller arc and won't reach the top of the arc where "h" is, so you know h > 4
-Only option for 4 < h < 6 is D
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Join the centre and the point on circumference, its length is 6 feet(radius)
Now look at the triangle formed, it is a right angled triangle, with hypotenues being 6, base 2.
Now hieght will be sqrt(36-4)= 4*sqrt(2)
Now look at the triangle formed, it is a right angled triangle, with hypotenues being 6, base 2.
Now hieght will be sqrt(36-4)= 4*sqrt(2)
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Since the radius is 6 feet, we can create a right triangle with hypotenuse of 6, leg of 2, and leg of h; thus, we have:
2^2 + h^2 = 6^2
4 + h^2 = 36
h^2 = 32
h = √16 x √2 = 4√2
Answer: D
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