If the Area of a Triangle ABC is given by the formula

Question

If the Area of a Triangle ABC is given by the formula 1/4 (a^2 + b^2), where a and b are the lengths of the two sides, then the angles of the triangle are?

A) 30, 60, 90
B) 60,40,80
C) 60,60,60
D) 90,45,45
E) None of the above

154238 · Accepted Answer

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EvaJager · Answer

The answer cannot be C: in this case, the triangle would be equilateral, so necessarily  a = b. 
Then  \frac{a^2+b^2}{4}=\frac{2a^2}{4}  while the area of the triangle would be  \frac{\sqrt{3}a^2}{4}. 
The two expressions are equal only if  a=0.

EvaJager · Answer

The triangle cannot be equilateral (see my previous post above if-the-area-of-a-triangle-abc-is-given-by-the-formula-140723.html#p1131446). We can eliminate answer B, as on the GMAT we don't have any means to test it. If the triangle is a right triangle, then a and b can be its two legs, or one is a leg and the other one is the hypotenuse. It is easier to test the case when a and b are legs. If the triangle is a right triangle with its two legs a and b , then a^2+b^2 is the square of the diameter of the circle in which this triangle is inscribed. \frac{a^2+b^2}{4} represents half of the area of the square inscribed in the same circle. Recall the formula for the area of a square being half of the diagonal squared, which in this case is \frac{a^2+b^2}{2} . It follows that the triangle is an isosceles right triangle. There is just one correct answer (this is a PS question), so we should stop checking other options. A cannot be the answer, in the expression for the area of the triangle, a factor of \sqrt{3} would necessarily appear (try to work out the two cases), but on the test, we should stop after finding the correct answer rather then continue to prove that the other options are not acceptable. Answer D.

vomhorizon · Answer

Thanks for pointing it out. I have removed it the OG...

MacFauz · Answer

a) Let the sides of the triangle be 1,sqrt(3),2, area = 0.5*1*sqrt(3) but according to formula, area = .25*(1+3) = .25*4. So A is wrong.

d) Let the sides of the triangle be 1,1 and sqrt(2), area = 0.5*1*1. According to formula, area = .25*(1+1) = 0.5. So answer is D

maddyboiler · Answer

I did it in this way. Not able to figure out. I was able to rule out all the other options. How is it D?
Here is what i did.

Now the question says that two sides are a and b, it doesnt talk about the 3rd side. In case of option D
Case 1(a,b,c) here let a = b and c be the hypotenuse 
ar = (1/2)base×height = (1/2)ab = (1/2)a²
area from the equation = (1/4)(a² + b²) = (1/4)(2a²) = (1/2)a²
both are equal

Case 2 (a,a,b)
so a² + a² = b² = 2a²
ar = (1/2)base×height = a²/2
ar from the equation = (1/4)(a² + b²) = (1/4)(a² + 2a²) = (3/4)a²
the above two are not equal.

Both the cases are contradicting

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If the Area of a Triangle ABC is given by the formula

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If the Area of a Triangle ABC is given by the formula

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