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

Question

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.

chetan2u · Answer

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.

TestPrepUnlimited · Answer

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.

Kinshook · Answer

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

Let the coordinate of point C be (x,y)

AB =  \sqrt{(1-5)^2 + (2-5)^2} = \sqrt{4^2 + 3^2} = 5

Let us find a point C(x,y) on the circumference of a circle with diameter AB such that area of triangle formed △ABC is to be 6.5 square units.

Radius of the circle = 5/2 = 2.5 units

Area of triangle △ABC = 1/2* base * height = 1/2*AB*h = 1/2 *5 *h = 6.5 
 h = 6.5/2.5 = 2.6 units

But hmax = 2.5 since h can not exceed radius of the circle

Not such triangle is possible

IMO A

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

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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