# Basic Geometry : 45/45/90 Right Isosceles Triangles

## Example Questions

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### Example Question #1 : Triangles

What is the length of the diagonal of a square with sides of ?

Explanation:

The diagonal cuts a square into two right triangles with the legs of the triangle being the sides of the square and the hypotenuse is the diagonal of the triangle.

Using Pythagorean Theorem we get:

or

Explanation:

### Example Question #1 : 45/45/90 Right Isosceles Triangles

A right triangle has two sides of length ; what is the correct formula for finding the length of the hypotenuse ?

Explanation:

The Pythagorean Theorem states that the sum of the squares of the sides of the triangle are equal to the square of the hypotenuse, summed up as  but for the case of right isoceles triangles where a and b are equal, it can be rewritten as:

,

which simplifies to

.

### Example Question #1 : Triangles

is a  triangle.

Using the Pythagorean Theorem, calculate the length of the hypotenuse of .

There is not enough information given to answer this question.

Explanation:

The Pythagorean Theorem holds that, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.  That is,

where  is the hypotenuse and  and  are the other two sides.

Here,  corresponds to , and, since this is a  triangle, the length of  equals the length of , so we know that .

Apply the Pythagorean Theorem.

So the length of the hypotenuse is .

### Example Question #1 : Triangles

The following image is not to scale.

Find the length of the hypotenuse of the right triangle:

Explanation:

Because the problem states that this the triangle is a right triangle and provides the length of the two legs, the third side (hypotenuse) can be calculated using the Pythagorean Theorem.

, where a and b can be arbitrarily chosen for either leg length and c represents the length of the hypotenuse.

but because the question requires to find the length of the hypotenuse, c cannot be a negative number. Therefore the final answer is +13.9.

### Example Question #1 : 45/45/90 Right Isosceles Triangles

The following image is not to scale.

Find the length of the hypotenuse of the  right triangle.

Explanation:

45/45/90 triangles are always isosceles. This means that two of the legs of the triangle are congruent. In the figure, it's indicates which two sides are congruent.

From here, we can find the length of the hypotenuse through the Pythagorean Theorem. We can confirm this because the problem has given us no angle measures to perform trig functions with.

c also equals -2√2, but because the hypotenuse is a physical length, the value cannot be negative distance.

### Example Question #7 : Triangles

Square  has a side length of . What is the length of its diagonal?

Cannot be determined from the information provided

Explanation:

The answer can be found two different ways. The first step is to realize that this is really a triangle question, even though it starts with a square. By drawing the square out and adding the diagonal, you can see that you form two right triangles. Furthermore, the diagonal bisects two ninety-degree angles, thereby making the resulting triangles a  triangle.

From here you can go one of two ways: using the Pythagorean Theorem to find the diagonal, or recognizing the triangle as a  triangle.

1) Using the Pythagorean Theorem

Once you recognize the right triangle in this question, you can begin to use the Pythagorean Theorem. Remember the formula: , where  and  are the lengths of the legs of the triangle, and  is the length of the triangle's hypotenuse.

In this case, . We can substitute these values into the equation and then solve for , the hypotenuse of the triangle and the diagonal of the square:

The length of the diagonal is .

2) Using Properties of  Triangles

The second approach relies on recognizing a  triangle. Although one could solve this rather easily with Pythagorean Theorem, the following method could be faster.

triangles have side length ratios of , where  represents the side lengths of the triangle's legs and  represents the length of the hypotenuse.

In this case,  because it is the side length of our square and the triangles formed by the square's diagonal.

Therefore, using the  triangle ratios, we have  for the hypotenuse of our triangle, which is also the diagonal of our square.

### Example Question #8 : Triangles

What is the length of the hypotenuse of an isosceles right triangle with an area of ?

Explanation:

Recall that an isosceles right triangle is also a  triangle. It has sides that appear as follows:

Therefore, the area of the triangle is:

, since the base and the height are the same.

For our data, this means:

Solving for , you get:

So, your triangle looks like this:

Now, you can solve this with a ratio and easily find that it is .  You also can use the Pythagorean Theorem. To do the latter, it is:

Now, just do your math carefully:

That is a weird kind of factoring, but it makes sense if you distribute back into the group. This means you can simplify:

### Example Question #9 : Triangles

When the sun shines on a  pole, it leaves a shadow on the ground that is also . What is the distance from the top of the pole to the end of its shadow?

Explanation:

The pole and its shadow make a right angle. Because they are the same length, they form an isosceles right triangle (45/45/90). We can use the Pythagorean Theorem to find the hypotenuse. . In this case, . Therefore, we do . So

### Example Question #10 : Triangles

Find the length of the hypotenuse.