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OA is C.
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Length of the diameter
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Not quite sure how to tackle this one.
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OA is C.
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OA is C.
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Rahul@gurome
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Given: Arc XYZ forms semicircle => Triangle XYZ is right-angled.
=> (XY)² + (YZ)² = (XZ)² = (r + s)² ................................................. (i)
Now in the two small right-angled triangles:
Combining (i) and (ii) : (r + s)² = (32 + r² + s²) => rs = 16
We need to find length of XZ, i.e. (r + s).
Statement 1: r = 8
Clearly s = 2 => XZ = (r + s) = (8 + 2) = 10
Sufficient.
Statement 2: s = 2
Clearly r = 8 => XZ = (r + s) = (8 + 2) = 10
Sufficient.
The correct answer is D.
=> (XY)² + (YZ)² = (XZ)² = (r + s)² ................................................. (i)
Now in the two small right-angled triangles:
- 1. (4)² + (r)² = (YZ)²
2. (4)² + (s)² = (XY)²
Combining (i) and (ii) : (r + s)² = (32 + r² + s²) => rs = 16
We need to find length of XZ, i.e. (r + s).
Statement 1: r = 8
Clearly s = 2 => XZ = (r + s) = (8 + 2) = 10
Sufficient.
Statement 2: s = 2
Clearly r = 8 => XZ = (r + s) = (8 + 2) = 10
Sufficient.
The correct answer is D.
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In a semicircle, an inscribed angle that intercepts the diameter is 90 degrees. Thus, ∠XYZ is a right angle, making triangle XYZ a right triangle. The line drawn through the right angle at Y is a height of triangle XYZ because it is perpendicular to line XZ (which we can consider the base).
In any right triangle, a height drawn through the right angle to the opposite side creates 3 similar triangles.
Thus, the small right triangle at the bottom, the larger right triangle at the top, and right triangle XYZ are all similar.
The corresponding sides of similar triangles must yield the same proportion.
In the small right triangle at the bottom, the shorter leg = s, the longer leg = 4.
In the larger right triangle at the top, the shorter leg = 4, the longer leg = r.
Since shorter leg:longer leg must be the same for each triangle, we get:
s/4 = 4/r.
Statement 1:
If r=8, then s/4 = 4/8, and s=2.
Diameter = r+s = 8+2 = 10.
Sufficient.
Statement 2:
If s=2, then 2/4 = 4/r, and r=8.
Diameter = r+s = 8+2 = 10.
Sufficient.
The correct answer is D.
Please confirm the OA. C is not the correct answer.
For another discussion of the 3 similar triangles formed when a height is drawn through the right angle of a right triangle, please check the following thread:
https://www.beatthegmat.com/geo-question ... tml#325797
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Statement 1 makes it clear that r=8 is the longer leg in the top right triangle, since the other leg has a length of 4.[email protected] wrote:Mitch:
How did you make the determination of which side is shorter or longer?
Statement 2 makes it clear that s=2 is the shorter leg in the bottom right triangle, since the other leg has a length of 4.
Please note that the two values could be reversed. If r=2, then it would be the shorter leg in the top right triangle. If s=8, then it would be the longer leg in the bottom right triangle.
Either way, the proportion will be the same: s/4 = 4/r.
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Equation (i) : (XY)² + (YZ)² = (r + s)²HPengineer wrote:lost you here
Combining (i) and (ii) : (r + s)² = (32 + r² + s²) => rs = 16
Equation (ii) : (XY)² + (YZ)² = (32 + r² + s²)
LHS of both of the equations are same, thus RHS must be same also.
Thus, (r + s)² = (32 + r² + s²)
=> (r² + 2rs + s²) = (32 + r² + s²)
=> 2rs = 32
=> rs = 16
Hope it is clear now.
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I still did not understand! How do we know if s or r are the longer or shorter legs in each of the triangles, because when you consider let say the first statement, we do not know that s equals to 2, so that we can use the s/4 = 4/r, do we? Since I do not know s, I could have said 4/s = 4/r and think that s is greater than 4 from the info provided in the first statement. Please, can you help me clarify this? Thanks in advance!GMATGuruNY wrote:Statement 1 makes it clear that r=8 is the longer leg in the top right triangle, since the other leg has a length of 4.[email protected] wrote:Mitch:
How did you make the determination of which side is shorter or longer?
Statement 2 makes it clear that s=2 is the shorter leg in the bottom right triangle, since the other leg has a length of 4.
Please note that the two values could be reversed. If r=2, then it would be the shorter leg in the top right triangle. If s=8, then it would be the longer leg in the bottom right triangle.
Either way, the proportion will be the same: s/4 = 4/r.
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I've assigned variables to the angles so that the corresponding sides can be identified more easily.finance wrote:I still did not understand! How do we know if s or r are the longer or shorter legs in each of the triangles, because when you consider let say the first statement, we do not know that s equals to 2, so that we can use the s/4 = 4/r, do we? Since I do not know s, I could have said 4/s = 4/r and think that s is greater than 4 from the info provided in the first statement. Please, can you help me clarify this? Thanks in advance!GMATGuruNY wrote:Statement 1 makes it clear that r=8 is the longer leg in the top right triangle, since the other leg has a length of 4.[email protected] wrote:Mitch:
How did you make the determination of which side is shorter or longer?
Statement 2 makes it clear that s=2 is the shorter leg in the bottom right triangle, since the other leg has a length of 4.
Please note that the two values could be reversed. If r=2, then it would be the shorter leg in the top right triangle. If s=8, then it would be the longer leg in the bottom right triangle.
Either way, the proportion will be the same: s/4 = 4/r.
In each triangle, (side opposite a)/(side opposite b) must yield the same ratio:
Thus, s/4 = 4/r.
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r^2+2rs+s^2 = r^2+16 + s^2 + 16
thus for,
a r=8 we get after solving s=2
b s=2 we get after solving r=8.
hence D it is.
thus for,
a r=8 we get after solving s=2
b s=2 we get after solving r=8.
hence D it is.
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