Find the perimeter.

Explanation

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:

Screen shot 2016 07 07 at 11.31.44 am

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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

Screen shot 2016 07 07 at 12.58.14 pm

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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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.

Recall how to find the perimeter of a triangle:

$\text{Perimeter}=a+b+c$

The given triangle has of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of side , rewrite the equation.

$b=\sqrt{c^2-a^2}$

Plug in the values of and .

$b=\sqrt{12^2-8^2}=\sqrt{80}$

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to decimal places.

$\text{Perimeter}=12+8+\sqrt{80}=28.94$

2

Find the perimeter.

Explanation

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:

Screen shot 2016 07 07 at 11.31.44 am

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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

Screen shot 2016 07 07 at 12.58.14 pm

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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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.

Recall how to find the perimeter of a triangle:

$\text{Perimeter}=a+b+c$

The given triangle has of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of side , rewrite the equation.

$b=\sqrt{c^2-a^2}$

Plug in the values of and .

$b=\sqrt{16^2-13^2}=\sqrt{87}$

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to decimal places.

$\text{Perimeter}=13+16+\sqrt{87}=38.33$

3

Find the perimeter.

Explanation

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:

Screen shot 2016 07 07 at 11.31.44 am

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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

Screen shot 2016 07 07 at 12.58.14 pm

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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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.

Recall how to find the perimeter of a triangle:

$\text{Perimeter}=a+b+c$

The given triangle has of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of the hypotenuse, , rewrite the equation.

$c=\sqrt{a^2+b^2}$

Plug in the values of and .

$c=\sqrt{5^2+9^2}=\sqrt{106}$

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to decimal places.

$\text{Perimeter}=5+9+\sqrt{106}=24.30$

4

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

True

False

Explanation

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is $60^{\circ }$ ). It follows that all angles of both triangles measure $60^{\circ }$ .

Specifically, $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , making the correct answer "true".

5

In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?

2√5

11

10√2

15

6√2

Explanation

Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100√2=c. 10√2=c

6

Find the perimeter.

Explanation

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:

Screen shot 2016 07 07 at 11.31.44 am

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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

Screen shot 2016 07 07 at 12.58.14 pm

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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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.

Recall how to find the perimeter of a triangle:

$\text{Perimeter}=a+b+c$

The given triangle has of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of side , rewrite the equation.

$b=\sqrt{c^2-a^2}$

Plug in the values of and .

$b=\sqrt{16^2-13^2}=\sqrt{87}$

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to decimal places.

$\text{Perimeter}=13+16+\sqrt{87}=38.33$

7

Find the perimeter of this right triangle, rounding to the nearest hundredths place.

Right triangle perim 2

$\small 16.74$

$\small 5.74$

$\small 19.06$

$\small 18.28$

Explanation

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:

Screen shot 2016 07 07 at 11.31.44 am

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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

Screen shot 2016 07 07 at 12.58.14 pm

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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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 find the perimeter, we need to know all three side lengths. To discover the missing leg, we can use the Pythagorean theorem. Note that we actually know the hypotenuse, , so we're solving for either or , one of the legs:

$\small 4^2 + b^2 = 7^2$

$\small 16 + b^2 = 49$ subtract 16 from both sides

$\small b^2 = 33$ take the square root

$\small b \approx 5.74$

Now that we have our missing side, we can find the perimeter by adding all three sides together:

$\textup{Perimeter}=a+b+c$

$\textup{Perimeter}=4+5.74 + 7 = 16.74$

8

Find the perimeter.

Explanation

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:

Screen shot 2016 07 07 at 11.31.44 am

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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

Screen shot 2016 07 07 at 10.13.54 am

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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

Screen shot 2016 07 07 at 12.58.14 pm

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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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 find the perimeter, we must first find the length of the hypotenuse of the right triangle.

Use Pythagorean's theorem to find the length of the hypotenuse:

$c=\sqrt{a^2+b^2}$

Plug in the values of the lengths of the legs of the given triangle.

$c=\sqrt{3^2+20^2}=\sqrt{409}$

Now, recall how to find the perimeter of a triangle:

$\text{Perimeter}=a+b+c$

Plug in all the values of the sides of the triangle to find the perimeter.

$\text{Perimeter}=3+20+\sqrt{409}=43.22$

Make sure to round to places after the decimal.

9

Find the perimeter.

Explanation

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:

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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:

$\textup{Perimeter}=a+b+c$

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 $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

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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.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

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.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

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.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

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:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

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:

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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:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

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.