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Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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12 00:00

Difficulty:   65% (hard)

Question Stats: 65% (02:58) correct 35% (02:44) wrong based on 199 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

Kudos for a correct solution.

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Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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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.
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Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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the answer is B

the circle will form tringle that have absic =8 and hight= 3

so the area is 1/2*8*3=12
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Manager  B
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Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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Center of the circle =(5,3) & Radius = 5; circle touches y axis at (0,3), and X axis at (1,0) & (9,0) (with help of radius y intercept we can find x intercept using Pythagorean theorem ).
Therefore Area of triangle = 3*8 /2 = 12
Therefore ans is B

Thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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Bunuel wrote:
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

Kudos for a correct solution.

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 3730 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.

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Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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Quote:
What is the area of the triangle formed by these three intercepts

Shouldn't this be rephrased as "What is the area of the triangle formed by the x-inercepts and the centre?
Retired Moderator Joined: 06 Jul 2014
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GMAT 1: 660 Q48 V33 GMAT 2: 740 Q50 V40 Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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StrikerT wrote:
Quote:
What is the area of the triangle formed by these three intercepts

Shouldn't this be rephrased as "What is the area of the triangle formed by the x-inercepts and the centre?

No, because question asks about about area of triangle with vertexes at coordinates: (0, 3), (1, 0) and (9, 0)

On the picture we see height of triangle that drew from point (5, 3) because this is height and doesn't matter from which point we will draw it: from (0, 3) or from (5, 3) it's still height of this triangle.
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Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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Harley1980 wrote:
StrikerT wrote:
Quote:
What is the area of the triangle formed by these three intercepts

Shouldn't this be rephrased as "What is the area of the triangle formed by the x-inercepts and the centre?

No, because question asks about about area of triangle with vertexes at coordinates: (0, 3), (1, 0) and (9, 0)

On the picture we see height of triangle that drew from point (5, 3) because this is height and doesn't matter from which point we will draw it: from (0, 3) or from (5, 3) it's still height of this triangle.

Ah. How can I miss that. Thanks.
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GMAT 1: 630 Q49 V27 Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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Bunuel wrote:
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

Kudos for a correct solution.

At Y=0 ,
(x-5)^2 + 9 = 25 ; X-5 = +-4; X=9,1
At X=0
25+(Y-3)^2 = 25 ; Y=3.

area of triangle = 1/2 * 3 * (9-1) = 12 .
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Re: Point W = (5, 3). Circle J has a center at point W and radius of r =  [#permalink]

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