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In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be

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LamboWalker
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In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be [#permalink] New post  03 Sep 2021, 11:06
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In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be 6.5 square units, then the possible number of points for C is

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be answered based on the given information.
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Re: In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be [#permalink] New post  03 Sep 2021, 20:38
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LamboWalker wrote:
In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be 6.5 square units, then the possible number of points for C is

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be answered based on the given information.



This means the hypotenuse is AB =\(\sqrt{(5-1)^2+(5-2)^2}=\sqrt{16+9}=5\)

When will the area be maximum for a triangle with hypotenuse?
It will be when the triangle is isosceles triangle so each side = \(\frac{5}{\sqrt{2}}\)
Max area = \(\frac{1}{2}* \frac{5}{\sqrt{2}}* \frac{5}{\sqrt{2}}=\frac{25}{4}=6.25\)

Thus, there will be no triangle with hypotenuse 5, that can have area more than 6.25 .

A

But it is slightly illogical to say that area is 6.5, when there are no such triangles possible. Could have been worded a bit better.
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Re: In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be [#permalink] New post  05 Sep 2021, 07:30
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LamboWalker wrote:
In triangle ABC, A=(1,2) B=(5,5), ∠ACB=90°. If area of △ABC is to be 6.5 square units, then the possible number of points for C is

(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be answered based on the given information.


Note ∠C must be the right angle. Then AB must be the hypotenuse. Using the Pythagorean theorem we can find AB = \(\sqrt{4^2 + 3^2} = 5\).

If we wanted to maximize the area of ABC, then ABC would be a 45-45-90 triangle, and the height on the AB side would be 2.5. This results in a maximum area of 2.5*2.5 = 6.25. Hence there are no triangles that can have an area of 6.5.

Ans: A

Notes there are 2 triangles with a target area of 6.25, and 4 triangles with a target area less than 6.25.
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Re: In triangle ABC, A=(1,2) B=(5,5), ACB=90°. If area of ABC is to be [#permalink] New post  06 May 2023, 08:34
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Re: In triangle ABC, A=(1,2) B=(5,5), ACB=90°. If area of ABC is to be [#permalink]
06 May 2023, 08:34
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