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10 Mar 2015, 06:14
Kudos
Bookmarks
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
64% (02:59) correct
36%
(02:46)
wrong
based on 408
sessions
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Point W = (5, 3). Circle J has a center at point W and radius of r = 5. This circle intersects the y-axis at one intercept and the x-axis at two intercepts. What is the area of the triangle formed by these three intercepts?
(A) 7.5
(B) 12
(C) 15
(D) 24
(E) 30
(A) 7.5
(B) 12
(C) 15
(D) 24
(E) 30
10 Mar 2015, 06:33
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Answer is B.
Circle centered at (5,3), with radius 5 will touch Y-axis at (0,3) point. So, the height of the triangle is 3.
For other 2 vertices of the triangle that lie on x-axis, we need to solve the circle equation (x-5)^2 + (y-3)^2 = 25 for y=0.
x = 9 and 1, which gives base length = 9-1 = 8. Now from area of triangle formula, area of triangle = 12.
Circle centered at (5,3), with radius 5 will touch Y-axis at (0,3) point. So, the height of the triangle is 3.
For other 2 vertices of the triangle that lie on x-axis, we need to solve the circle equation (x-5)^2 + (y-3)^2 = 25 for y=0.
x = 9 and 1, which gives base length = 9-1 = 8. Now from area of triangle formula, area of triangle = 12.
15 Mar 2015, 21:35
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Bunuel
MAGOOSH OFFICIAL SOLUTION:
Well, if we go 5 units to the left of (5, 3), we’re at (0, 3): that’s the single y-intercept of the circle.
Now, think about the two x-intercepts. Each one is a diagonal distance of r = 5 from (5, 3), and if we may a right triangle on either side, the vertical leg is the distance from (5, 3) straight down to the x-axis, which of course is 3.
Attachment:
cgpq_img7.png [ 22.39 KiB | Viewed 12216 times ]
Those two purple triangles must be 3-4-5 triangles, which means each one has a base of 4, and the distance between the two of them is 8. One is at (1, 0) and the other is at (9, 0).
Now, think about the triangle formed by these three intercepts. The base, from (1,0) to (9, 0) is 8, and while the third vertex is off-center, that doesn’t matter — the height is h = 3. A = (0.5)bh = (0.5)(8)(3) = 12.
Answer = (B)
