Which of the following is sufficient to determine the area of isosceles right triangle XYZ (not shown)?
I. The height of the triangle
II. The perimeter of the triangle
III. Any one side of the triangle
- (A) I only
- (B) I & III
- (C) II & III
- (D) II only
- (E) I, II, and III
Even though we don’t have a figure, we have been given a lot of information. Triangle XYZ is an isosceles right triangle. To translate that into simple terms: two sides are equal, and these two sides are the shortest sides. The longest side, the hypotenuse, will be opposite from the 90 degree angle. This type of triangle is known as a 45-45-90 triangle. The ratio of its sides is x: x: x?2, in which ‘x’ stands for the two equal sides and x?2 is the longest side (remember that ?2 equals roughly 1.4). So the longest side is 1.4 times the two shortest sides.
Let’s deal with the (I) first. If we know the height of the triangle, we know ‘x.’ The area of the triangle is (x)(x)/2, or x^2/2. Remember that height and base meet at a right angle, as do the two equal sides of a 45-45-90 triangle. If we know the height, we can solve for ‘x’, so (1) is sufficient.
For (II), we can simply add up the perimeter in terms of x, which gives us 2x + x?2. If we know the perimeter, we can solve for ‘x’. Once we know ‘x’, we can find the area. Therefore (II) is sufficient.
Finally, (III) is also sufficient, because by knowing any one side of the triangle, we can solve for ‘x’. For instance, if we know that the hypotenuse is equal to 4, we just solve for ‘x’: x?2 = 4, x = 2?2. Now that we know ‘x’, we can find the area. Therefore, the answer is (E).
This question was written by Chris Lele, resident GRE expert at Magoosh. Sign up for your free trial now to access to even more questions!