In a circle with center C and radius 62 cm, PQ and SR are two parallel chords separated by one of the diameters. If ∠PQC=45∘,andtheratiooftheperpendiculardistanceofPQandSRfromCis3:2 , then the area, in sq. cm, of the quadrilateral PQRS is
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Official Correct Answer: C. First, let's parse the problem:
We have a circle with center C and radius 6√2 \text{ cm}. There are two parallel chords, PQ and SR, separated by one of the diameters. The angle ∠PQC is 45 degrees, and the ratio of the perpendicular distances of PQ and SR from C is 3:2. We need to find the area of quadrilateral PQRS.
Hmm, okay. Consider this. There's a circle, center C. Two parallel chords, PQ and SR, which are on opposite sides of the center since they're separated by a diameter. The angle at PQC is 45 degrees, which is probably in triangle PQC. The distances from C to PQ and SR are in a 3:2 ratio.
I think I should start by recalling some circle theorems and properties related to chords and distances from the center.
**Core Concept & Formula:**
The perpendicular distance from the center of a circle to a chord can be found using the formula:
$$
d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2}
$$
where \( d \) is the distance, \( r \) is the radius, and \( l \) is the length of the chord.
Also, for two parallel chords, the distance between them can be found by adding or subtracting their individual distances from the center, depending on their positions.
Given that PQ and SR are separated by a diameter, the distance between them should be twice the radius, which is 12√2 \text{ cm}. Wait, no, that's the length of the diameter. But the distance between the two chords is the sum of their distances from the center if they are on opposite sides. So if their distances are in the ratio 3:2, let's denote them as 3k and 2k. Then the total distance between them is 3k + 2k = 5k. But the problem says they are separated by a diameter, which is 12√2 \text{ cm}. So 5k = 12√2, which gives k = (12√2)/5.
Let me make sure. If the two chords are on opposite sides of the center, the distance between them is indeed the sum of their distances from the center. So yes, 3k + 2k = 5k = 12√2. So k = (12√2)/5.
Therefore, the distance of PQ from C is 3k = (36√2)/5 \text{ cm}, and the distance of SR from C is 2k = (24√2)/5 \text{ cm}.
But The radius is 6√2 \text{ cm}, which is approximately 8.485 \text{ cm}. If the distance of PQ from C is (36√2)/5, which is approximately (36*1.414)/5 ≈ 50.904/5 ≈ 10.18 \text{ cm}. But the radius is only 6√2 ≈ 8.485 \text{ cm}. That can't be possible because the distance from the center can't exceed the radius. So I must have made a mistake here.
Ah, right! The distance from the center to the chord can't exceed the radius. So my initial assumption that the distance between the chords is equal to the diameter is incorrect. Wait, no, the distance between the chords is equal to the diameter? No, the distance between the chords is the length of the diameter, which is 12√2 \text{ cm}. But that would mean the sum of their distances from the center is 12√2 \text{ cm}. But if each distance is less than the radius, which is 6√2 \text{ cm}, then their sum can't exceed 12√2 \text{ cm}, which is exactly the diameter. So that makes sense.
Wait, but 3k + 2k = 5k = 12√2, so k = (12√2)/5. Then, the distance of PQ from C is 3k = (36√2)/5 ≈ 10.18 \text{ cm}, which is greater than the radius 6√2 ≈ 8.485 \text{ cm}. That's impossible because the distance from the center to the chord can't be more than the radius. So I must have messed up the ratio.
Wait, maybe the ratio is 3:2, but which one is closer? If PQ is farther from the center, then its distance is 3k, and SR is closer with 2k. But if 3k > 2k, then 3k must be less than the radius. So 3k < 6√2. Therefore, k < 2√2. Then 5k = 12√2, so k = (12√2)/5 ≈ 3.394 \text{ cm}. Then 3k ≈ 10.18 \text{ cm}, which is still more than 6√2 ≈ 8.485 \text{ cm}. Hmm, that's a problem.
Wait, maybe the ratio is 3:2, but the distances are measured from the center, so if the chords are on opposite sides, the total distance between them is 3k + 2k = 5k. But the distance between them is the diameter, which is 12√2. So 5k = 12√2, so k = (12√2)/5. Then, the distances are 3k = (36√2)/5 ≈ 10.18 \text{ cm} and 2k = (24√2)/5 ≈ 6.788 \text{ cm}. But 10.18 \text{ cm} is greater than the radius, which is 6√2 ≈ 8.485 \text{ cm}. That's impossible because the distance from the center to the chord can't exceed the radius.
So, I must have misunderstood the problem. Maybe the distance between the chords is not the diameter, but the diameter is one of the lines separating them. Wait, the problem says "separated by one of the diameters." So the distance between PQ and SR is equal to the length of the diameter, which is 12√2 \text{ cm}. So the sum of their distances from the center is 12√2 \text{ cm}.
But as we saw, that leads to a contradiction because 3k + 2k = 5k = 12√2, so k = (12√2)/5, and 3k = (36√2)/5 ≈ 10.18 \text{ cm} > 6√2 \text{ cm}. So that's impossible.
Wait, maybe the distance between the chords is the diameter, but the distance from the center to each chord is measured in the same direction, so the total