GMAT Math: Area of a Triangle

January 4, 2012

As you may remember from high school, \(A = \frac{1}{2}bh\) , where b is the base and h is the height.

If you are having trouble remembering this, simply remember that a rectangle has an area of \(A = bh\), and that a triangle is half a rectangle.

Practice Question: Using the Area Formula

isosceles right triangle & square with equal area

The figure on the left is an isosceles right triangle, and the figure on the right is a square of length 3. Find the value of b.

(1): b is the length of the diagonal of the square. (2): the triangle and the square have the same area. Answer Choices: A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient. B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are not sufficient.

Answer and Explanation

Statement (1) says that b is the length of the diagonal of the square. We know the side of the square is 3, so we could find the length of its diagonal with the ubiquitous Pythagorean Theorem. Therefore, we would know b. Statement (1) is sufficient by itself. Statement (2) says that the triangle and the square have equal area. Well, we know the area of the square is 9. The triangle has a base of b, and because we know it’s isosceles, we know its height is also b. It’s area, \(\frac{1}{2}b^2\), must equal 9. That means, we can solve for b. Statement (2) is also sufficient by itself. Answer: D

Author

Mike MᶜGarry

Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as “member of the month” for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike’s Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.

View all posts

Share16

Comments

8 responses to “GMAT Math: Area of a Triangle”

viji

June 30, 2016

Can you please explain, three altitudes of triangle are 12,9.6, 16. Find the area of triangle? how to solve this please explain.

Reply
1. Magoosh Test Prep Expert
  
  July 21, 2016
  
  Hi Viji,
  
  An altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base (the opposite side of the triangle). As explained on this Math Forum page, we can find the area of a triangle given its three altitudes by rewriting Heron’s formula:
  
  A = 1/sqrt[(1/ha + 1/hb + 1/hc)*(-1/ha + 1/hb + 1/hc)*(1/ha – 1/hb + 1/hc)*(1/ha + 1/hb – 1/hc)]
  
  where A is the area of the triangle and ha, hb, and hc are the altitudes of the triangle.
  
  So, if we plug in ha=12, hb=9.6, and hc=16, we find that the area is
  
  A = 1/sqrt[(1/12 + 1/9.6 + 1/16)*(-1/12 + 1/9.6 + 1/16)*(1/12 – 1/9.6 + 1/16)*(1/12 + 1/9.6 – 1/16)]
  A = 96
  
  Hope this helps!
  
  Reply
Etin Benin

August 18, 2015

From the diagram of the triangle, we do not necessarily know that b is one of the legs of the triangle. “b” could be the hypotenuse, because the <90 was not indicated. Then, statement 2 would be insufficient.

Reply
1. Ingrida
  
  February 9, 2016
  
  I also agree with your statement!
  The answer should be A
  
  Reply
2. S
  
  February 20, 2021
  
  The triangle figure shows which angle is 90deg. The sum of the other two angles therefore should be 90deg, which would imply that they are the smaller angles since you cannot have a triangle with more than one angle with a measure of 90deg or larger. Hence ‘b’ is a leg and the side opposite the 90deg angle is the longest side and hence the hypotenuse. [Unless the right angle indication was added after the initial comment was posted]
  
  Reply
  1. Magoosh Test Prep Expert
    
    March 11, 2021
    
    This is correct! The figure marks which angle is the right angle, and the side opposite it (which we see as a diagonal line) must be the hypotenuse. The question tells us that this is an isosceles right triangle. Therefore, we know for sure that both the horizontal and the vertical side [the two sides adjacent to the right angle] are legs, and they both have a length of b.
    
    More specifically: an “isosceles right triangle” has one 90 degree angle and two 45 degree angles. The side opposite the right angle is the hypotenuse. The two sides opposite the 45 degree angles are the legs, and they have the same length.
    
    Reply
Sandy Eastling

March 6, 2012

Where are the answer choices? Answer is “d” but I’m not seeing the choices.

Reply
1. Mike McGarry
  
  March 6, 2012
  
  Mr. Eastling: Underneath “Statement #1” and “Statement #2”, those five statements labeled (A) though (E) —those are the answer choices. Those are actually the standard answer choices for *all* Data Sufficiency questions — if those are not familiar, I definitely would recommend watching the Lesson Videos on DS in the near future. Please let me know if you have any further questions.
  Mike 🙂
  
  Reply