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# The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle

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Math Expert
Joined: 02 Sep 2009
Posts: 54543
The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle  [#permalink]

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28 Jul 2017, 09:01
2
7
00:00

Difficulty:

55% (hard)

Question Stats:

74% (03:01) correct 26% (03:16) wrong based on 77 sessions

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The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle. If angle ABC = 90°, what is the area of triangle ABC?

A. 102
B. 120
C. 132
D. 144
E. 256

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Joined: 03 Jun 2017
Posts: 15
Re: The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle  [#permalink]

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28 Jul 2017, 09:14
Given that triangle ABC is right angled at B, we know that the x coordinate of B and the x coordinate of C should be equal.

Equating both, we get a equals 5.5. Substitute into the points and the base comes out to be 17 units while the height is 12.

Area is 17*12/2 which is 102.

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Re: The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle  [#permalink]

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29 Jul 2017, 01:54
1
We know that since Y coordinate of point B and y coordinate of point C is equal (Triangle ABC is 90 at B)

Therefore equating the the y coordinates of both the points B and C

i.e 4a - 5 = 2a + 6

We get, a = 11/2

Substituting the value of a in the coordinate points. Hence the points are now A (0,0), B (0,17) and C(12,17)

Area of triangle ABC = 1/2 x 12 x 17 = 102

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Re: The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle  [#permalink]

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29 Jul 2017, 02:53
1
Bunuel wrote:
The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle. If angle ABC = 90°, what is the area of triangle ABC?

A. 102
B. 120
C. 132
D. 144
E. 256

ABC is a right angle triangle, right angled at B. ABC = 90°
Now A(0,0) and B (0, 4a-5) represents Y-axis..
So, BC must be perpendicular to Y - axis
i.e. BC must be parallel to X-axis

So. 4a-5=2a+6
-> 2a = 11
-> a = 5.5

So, 4a -5 = 17
2a+1 = 12
2a+6 = 17

So, the points are A(0,0) , B(0,17) and C(12,17)
AB = 17
BC = 12
Area (tria ABC) = 1/2*17*12 = 17 * 6 = 102

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Re: The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle  [#permalink]

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29 Jul 2017, 10:02
3
Bunuel wrote:
The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle. If angle ABC = 90°, what is the area of triangle ABC?

A. 102
B. 120
C. 132
D. 144
E. 256

• A very interesting testing our skill of visualization.
• A is the origin and B lies on the y-axis.
o If ABC is a right-angled triangle, that mean angle B = 90 degrees.
o Also, line BC is a parallel to the x-axis.
 Thus, the y-cooridante of B and C would be the same.
• $$4a – 5 = 2a + 6$$
• $$2a = 11$$
• $$a = 5.5$$
• The area of the triangle $$= \frac{1}{2} * AB * BC = \frac{1}{2} * (4a -5) * (2a+1)$$
o We know that $$a = 5.5$$, substituting the value in the equation above, we get –
 $$Area = \frac{1}{2} * (22-5)*12$$
 $$Area = 102$$
• And the correct answer is Option A.

Thanks,
Saquib
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Re: The points A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6) form a triangle   [#permalink] 29 Jul 2017, 10:02
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