|
As previously noted, an isosceles triangle is one in which two sides are equal. The two angles opposite those two sides will also be equal. The most important isosceles triangle on the GRE is the isosceles right triangle.
An isosceles right triangle has one 90° angle (opposite the hypotenuse) and two 45° angles (opposite the two equal legs). This triangle is called the 45–45–90 triangle.
The lengths of the legs of every 45–45–90 triangle have a specific ratio, which you must memorize:
|
|
What does it mean that the sides of a 45–45–90 triangle are in a
ratio? It doesn’t mean that they are actually 1, 1, and
(although that’s a possibility). It means that the sides are some multiple of
. For instance, they could be 2, 2, and
, or 5.5, 5.5, and
. In the last two cases, the number you multiplied the ratio by—either 2 or 5.5—is called the multiplier. Using a multiplier of 2 has the same effect as doubling a recipe—each of the ingredients gets doubled. Of course, you can also triple a recipe or multiply it by any other number, even a fraction. Try this problem:
If the length of side AB is 5, what are the lengths of sides BC and AC?
Because AB is 5, use the ratio
for sides AB : AC : BC to determine that the multiplier x is 5. You then find that the sides of the triangle have lengths
. Therefore, the length of side AC = 5, and the length of side
. Using the same figure, though without the information from the previous question, review the following problem:
If the length of side BC is
, what are the lengths of sides AB and AC?
Because the hypotenuse BC is
, solve for 
.
Thus, the sides AB and AC are each equal to x, which is 3.
One reason that the 45–45–90 triangle is so important is that this triangle is exactly half of a square! That is, two 45–45–90 triangles put together make up a square. Thus, if you are given the diagonal of a square, you can use the 45–45–90 ratio to find the length of a side of the square.
What is the area of a square with diagonal of 6?
What is the diagonal of a square with an area of 25?