As you may remember from high school, , 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 , and that a triangle is half a rectangle.
Practice Question: Using the Area Formula
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.
Statement #1: b is the length of the diagonal of the square.
Statement #2: the triangle and the square have the same area.
(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
(D) Each statement alone is sufficient to answer the question.
(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.
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, , must equal 9. That means, we can solve for b. Statement #2 is also sufficient by itself.