45/45/90 Right Isosceles Triangles - ACT Math

Now, this is really your standard triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

For your data, this is:

Now, this is really your standard triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

For your data, this is:

Right isosceles triangles (also called "45-45-90 right triangles") are special shapes. In a plane, they are exactly half of a square, and their sides can therefore be expressed as a ratio equal to the sides of a square and the square's diagonal:

, where is the hypotenuse.

In this case, maps to , so to find the length of a side (so we can use the triangle area formula), just divide the hypotenuse by :

So, each side of the triangle is long. Now, just follow your formula for area of a triangle:

Thus, the triangle has an area of .

Recall that an isosceles right triangle is a triangle. That means that it looks like this:

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

However, since for our triangle, we know:

Now, we know that . Therefore, we can write:

Solving for , we get:

This is the length of the height of the triangle for the side that is not the hypotenuse.

Recall that an isosceles right triangle is a triangle. That means that it looks like this:

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

However, since for our triangle, we know:

Now, we know that . Therefore, we can write:

Solving for , we get:

However, be careful! Notice what the question asks: "What is its height that is correlative and perpendicular to this triangle's hypotenuse?" First, let's find the hypotenuse of the triangle. Recall your standard triangle:

Since one of your sides is , your hypotenuse is .

Okay, what you are actually looking for is in the following figure:

Therefore, since you know the area, you can say:

Solving, you get: .

Based on the information given, you know that your triangle looks as follows:

This is a triangle. Recall your standard triangle:

You can set up the following ratio between these two figures:

Now, the area of the triangle will merely be (since both the base and the height are ). For your data, this is:

To solve simply realize the hypotenuse of one of these triangles is of the form where s is side length. Thus, our answer is .

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 .

1. 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.

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:

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 .

To solve, simply use the Pythagorean Theorem.

Recall that an isosceles right triangle has two leg lengths that are equal.

Therefore, to solve for the hypotenuse let and in the Pythagorean Theorem.

Thus,

It's helpful to remember upon coming across a 45/45/90 triangle that it's a special right triangle. This means that its sides can easily be calculated by using a derived side ratio:

Here, represents the length of one of the legs of the 45/45/90 triangle, and represents the length of the triangle's hypotenuse. Two sides are denoted as congruent lengths () because this special triangle is actually an isosceles triangle. This goes back to the fact that two of its angles are congruent.

Therefore, using the side rules mentioned above, if , this problem can be resolved by solving for the value of :

Therefore, the length of one of the legs is 1.

If the hypotenuse of a 45-45-90 triangle is provided, its side length can only be one length, since the sides of all 45-45-90 triangles exist in a defined ratio of , where represents the length of one of the triangle's legs and represents the length of the triangle's hypotenuse. Using this method, you can set up a proportion and solve for the length of one of the triangle's sides:

Cross-multiply and solve for .

Rationalize the denominator.

You can also solve this problem using the Pythagorean Theorem.

In a 45-45-90 triangle, the side legs will be equal, so . Substitute for and rewrite the formula.

Substitute the provided length of the hypothenuse and solve for .

While the answer looks a little different from the result of our first method of solving this problem, the two represent the same value, just written in different ways.

1. Remember that this is a special right triangle where the ratio of the sides is:

In this case that makes it:

2. Find the perimeter by adding the side lengths together:

Remember that this is a special right triangle where the ratio of the sides is:

In this case that makes it:

Where is the length of the hypotenuse.

This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem giving us . If then .

Your right triangle is a triangle. It thus looks like this:

Now, you know that you also have a reference triangle for triangles. This is:

This means that you can set up a ratio to find . It would be:

Your triangle thus could be drawn like this:

Now, notice that you can rationalize the denominator of :

Thus, the perimeter of your figure is:

Recall that an isosceles right triangle is also a triangle. Your reference figure for such a shape is:

or

Now, you know that the area of a triangle is:

For this triangle, though, the base and height are the same. So it is:

Now, we have to be careful, given that our area contains . Let's use , for "side length":

Thus, . Now based on the reference figure above, you can easily see that your triangle is:

Therefore, your perimeter is:

The ratio of sides to hypotenuse of an isosceles right triangle is always . With this in mind, setting as our hypotenuse means we must have leg lengths equal to:

Since the perimeter has two of these legs, we just need to multiply this by and add the result to our hypothesis:

So, our perimeter in terms of is: