### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find The Perimeter Of A Right Triangle

In the figure below, right triangle has a hypotenuse of 6. If and , find the perimeter of the triangle .

**Possible Answers:**

**Correct answer:**

# How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

- When side lengths are given, add them together.
- Solve for a missing side using the Pythagorean theorem.
- If we know side-angle-side information, solve for the missing side using the Law of Cosines.

## Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

## Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the angle and they are labeled and . The side of the triangle that is opposite of the angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

Simplify.

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

Simplify.

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

Simplify.

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

## Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

## Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

First, we need to use the Pythagorean theorem to solve for .

Because we are dealing with a triangle, the only valid solution is because we can't have negative values.

After you have found , plug it in to find the perimeter. Remember to simplify all square roots!

### Example Question #1 : How To Find The Perimeter Of A Right Triangle

Find the perimeter of the triangle below.

**Possible Answers:**

**Correct answer:**

# How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

- When side lengths are given, add them together.
- Solve for a missing side using the Pythagorean theorem.
- If we know side-angle-side information, solve for the missing side using the Law of Cosines.

## Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

## Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the angle and they are labeled and . The side of the triangle that is opposite of the angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

Simplify.

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

Simplify.

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

Simplify.

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

## Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

## Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem. The perimeter of a triangle is simply the sum of its three sides. Our problem is that we only know two of the sides. The key for us is the fact that we have a right triangle (as indicated by the little box in the one angle). Knowing two sides of a right triangle and needing the third is a classic case for using the Pythagorean theorem. In simple (sort of), the Pythagorean theorem says that sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse.

Every right triangle has three sides and a right angle. The side across from the right angle (also the longest) is called the hypotenuse. The other two sides are each called legs. That means in our triangle, the side with length 17 is the hypotenuse, while the one with length 8 and the one we need to find are each legs.

What the Pythagorean theorem tells us is that if we square the lengths of our two legs and add those two numbers together, we get the same number as when we square the length of our hypotenuse. Since we don't know the length of our second leg, we can identify it with the variable .

This allows us to create the following algebraic equation:

which simplified becomes

To solve this equation, we first need to get the variable by itself, which can be done by subtracting 64 from both sides, giving us

From here, we simply take the square root of both sides.

Technically, would also be a square root of 225, but since a side of a triangle can only have a positive length, we'll stick with 15 as our answer.

But we aren't done yet. We now know the length of our missing side, but we still need to add the three side lengths together to find the perimeter.

Our answer is 40.

### Example Question #3 : How To Find The Perimeter Of A Right Triangle

Given that two sides of a right triangle are and and the hypotenuse is unknown, find the perimeter of the triangle.

**Possible Answers:**

**Correct answer:**

# How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

- When side lengths are given, add them together.
- Solve for a missing side using the Pythagorean theorem.
- If we know side-angle-side information, solve for the missing side using the Law of Cosines.

## Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

## Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the angle and they are labeled and . The side of the triangle that is opposite of the angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

Simplify.

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

Simplify.

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

Simplify.

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

## Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

## Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

In order to calculate the perimeter we need to find the length of the hypotenuse using the Pythagorean theorem.

Rearrange.

Substitute in known values.

Now that we have found the missing side, we can substitute the values into the perimeter formula and solve.

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