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Math Expert V
Joined: 02 Sep 2009
Posts: 58442
If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 80% (01:32) correct 20% (01:31) wrong based on 258 sessions

HideShow timer Statistics If the area of right triangle ABC is 30, what are the coordinates of point B ?

A. (9, 4)
B. (4, 9)
C. (3, 4)
D. (-3, 9)
E. (12, 4)

Attachment: Geometry_Q9Image.pg.jpg [ 6.81 KiB | Viewed 2846 times ]

Kudos for a correct solution.

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Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Bunuel wrote: If the area of right triangle ABC is 30, what are the coordinates of point B ?

A. (9, 4)
B. (4, 9)
C. (3, 4)
D. (-3, 9)

E. (12, 4)

Attachment:
Geometry_Q9Image.pg.jpg

Kudos for a correct solution.

Point B will have same Y co-ordinate as Point A has i.e. 4. Hence, Option A, C and E remaining and other two options B and D are ruled out

Area of Triangle ABC = (1/2)*AB*AC = 30 (Given)

But AC = 5 (Distance of Y co-ordinate from -1 to 4 as X co-ordinates of A and C are same)

i.e. Area of Triangle ABC = (1/2)*AB*5 = 30 i.e. AB = 12

Horizontal Distance of 12 from X co-ordinate of point A (-3) = -3+12 = +9

i.e. Co-ordinate of point B = (9, 4)

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Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Triangle ABC is right triangle. So the area of ABC will be equal to half the product of its legs.
Also, length of leg AC is (4 -(-1)) = 4 + 1 = 5 (We can do this because x-coordinate is the same)
=> 30 = 1/2 * AC * AB = 1/2 * 5 * AB
=>AB = 60/5 = 12.

We know that ABC is a right angled triangle. And one of its legs if parallel to y-axis.
So, the other leg must be parallel to x-axis and the y-coordinate of points A and B must be same.
x-coordinate of leg AB = -3 + 12 = 9
So, coordinates of point B are (x,y) = (9,4)

Ans. A
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GMAT 1: 760 Q51 V41 Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Segment AC has a length of 4-(-1)=5.

Given that ABC is a right triangle, (AB*AC)/2=30. Therefore AB*5=60, AB=12.

Since the X coordinate of A and C are the same, you know that the Y coordinate of A and B would be the same as well.

To get the length of AB, just take -3+12=9. Point B is located at point (9,4). The answer is A
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Posts: 58442
Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Bunuel wrote: If the area of right triangle ABC is 30, what are the coordinates of point B ?

A. (9, 4)
B. (4, 9)
C. (3, 4)
D. (-3, 9)
E. (12, 4)

Attachment:
Geometry_Q9Image.pg.jpg

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Since side (AC) is parallel to the y-axis (based on the fact that they have the same x-coordinate) and we know that angle BAC is a right angle, we can conclude that side (AB) is parallel to the x-axis and therefore, once we determine the length, it’s simply a matter of moving that distance horizontally to the right. To determine the length of that side, use the formula for the area of a triangle: 1/2*(base)(height) = area.

We know the area is 30 from given information, and can calculate that (AC) has a length of 5. Using h for the unknown, we can calculate 1/2(5)(h) = 30 and solve that h= 12. Since (AB) has a length of 12, simply add 12 to the x-coordinate for the coordinates of point B, (9, 4).
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Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Logically, we can get to the right answer without any calculations:

A. (9, 4)

y interceptor of point B must be 4, therefore, we can eliminate B. (4, 9) and D. (-3, 9).
We know that AC = 5, which means that AB must be equal to 12. We can eliminate C because it tells us the x-interceptor of point B is 3, and E because it tells us that x-interceptor of point B is 12, which is as well impossible. The only correct answer choice left is A.
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Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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Bunuel wrote: If the area of right triangle ABC is 30, what are the coordinates of point B ?

A. (9, 4)
B. (4, 9)
C. (3, 4)
D. (-3, 9)
E. (12, 4)

Attachment:
Geometry_Q9Image.pg.jpg

Kudos for a correct solution.

Quite Simply, we are given the area of the triangle, a right triangle, so we can just use algebra to solve for the base. We know that one side of the triangle is 5 as indicated by the difference in y coordinates -1 and 4 so using the formula for the area of the right triangle we can reverse engineer

5 x Base / 2 = 30
5 *Base = 60
Base = 12
On that same logic, because we know the x coordinate of a point directly across from point B ( has the same y value) we can just find the difference by taking the absolute value of the x coordinate and using algebra to solve

3 + x= 12
x= 9

More fundamentally, if you count 3 spots to the right from the vertice of the triangle (-3 , 4) you will be at (0,4) and if you move three spots to the right you would be at (9,4) which would give you a total length of twelve for the base.

Thus
A
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Re: If the area of right triangle ABC is 30, what are the coordinates of  [#permalink]

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_________________ Re: If the area of right triangle ABC is 30, what are the coordinates of   [#permalink] 30 Aug 2018, 02:55
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