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29 May 2023, 23:37
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D
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In the given figure, if A is the center of the circle, CD is a straight line, and BE = 6, then what is the radius of the circle?
(1) DE is equal to 5.
(2) AF is equal to \(\frac{7}{8}\)
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29 May 2023, 23:38
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ChandlerBong
Hi Experts, KarishmaB ThatDudeKnows,
Is there any simple approach to solving this question? The given explanation for this question seems a bit time-consuming.
Thanks in advance
30 May 2023, 00:31
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ChandlerBong
Not an expert, sharing my two cents if that helps. IMO by observation, we can solve this question relatively quickly.
We can start with statement 2, as it's the easier of the two statements.
Statement 2
(2) AF is equal to \(\frac{7}{8}\)
BE is a chord and AF is perpendicular to that chord. Hence, BF = FE = 3 cms.
Property: A perpendicular from the center of the circle, bisects the chord.
\(\triangle BAF\) is a right-angled triangle, and we know the value of two sides of the triangle. Hence, the value of the third side, AB, which also happens to be the radius of the circle, can be found. Hence, statement 2 is sufficient.
Note: This is a DS question, we don't actually need to find the value. Knowing the fact that the value can be found, is sufficient.
Eliminate A, C, and E.
Statement 1
(1) DE is equal to 5.
There are two ways to solve this -
1) Mathematical
\(\triangle DFE\) and \(\triangle AFE\) are right angled triangle
From the question stem, we know the value of FE = 3 units
AE = AD = radius = r
Therefore we can find the value of AF in terms of the radius
AF = \(\sqrt{r^2-9}\)
FD = \(\sqrt{r^2-9} + r \) --- (1)
In \(\triangle DFE\) we know the value of DE and FE, hence we can equate the value of the third side FD to equation 1
\(DE^2 = FD^2 + FE^2\)
\(FD = 4\)
\(\sqrt{r^2-9} + r = 4\)
\(\sqrt{r^2-9} = 4 - r\)
Square both sides of the equation, and we can obtain the value of 'r'. Hence, the statement is sufficient.
2) Observation
\(\triangle BDE\) is an elongated (stretched) version of \(\triangle ABE\). This means the original point A when moved in a straight line by a distance 'r' leads to point D. The sides will extend in a 'fixed proportion'. Hence, knowing two sides of the \(\triangle BDE\) can help us find the sides of \(\triangle ABE\). Statement 2 provides us with that information.
We don't necessarily have to find that ratio or the values, but it can be done if need be. Hence, this statement is sufficient.
Option D
