Calculating the Length of Side b

Oct 13, 2025 by windailys.com 21 views

Understanding the Problem

To find the perimeter and area of the triangle, we need to use the given information and apply trigonometric concepts. The problem provides us with a triangle with a side length of 5 inches and an angle of 59°. We will use this information to calculate the perimeter and area of the triangle.

Calculating the Perimeter

The perimeter of a triangle is the sum of the lengths of its sides. To find the perimeter, we need to calculate the lengths of the other two sides using the given information. We can use the Law of Sines to find the lengths of the other two sides.

Law of Sines

The Law of Sines states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively, the following equation holds:

a / sin(A) = b / sin(B) = c / sin(C)

We can use this equation to find the lengths of the other two sides.

Calculating the Length of Side b

Using the Law of Sines, we can set up the following equation:

5 / sin(59°) = b / sin(90° - 59°)

We can simplify this equation by using the fact that sin(90° - x) = cos(x). Therefore, we have:

5 / sin(59°) = b / cos(59°)

We can solve for b by multiplying both sides by cos(59°):

b = 5 * cos(59°)

Using a calculator, we can find the value of b:

b ≈ 5 * 0.54 b ≈ 2.7

Calculating the Length of Side c

We can use the Law of Sines again to find the length of side c:

5 / sin(59°) = c / sin(90° - 59°)

We can simplify this equation by using the fact that sin(90° - x) = cos(x). Therefore, we have:

5 / sin(59°) = c / cos(59°)

We can solve for c by multiplying both sides by cos(59°):

c = 5 * cos(59°)

Using a calculator, we can find the value of c:

c ≈ 5 * 0.54 c ≈ 2.7

Calculating the Perimeter

Now that we have the lengths of all three sides, we can calculate the perimeter:

Perimeter = a + b + c Perimeter = 5 + 2.7 + 2.7 Perimeter ≈ 10.4

Calculating the Area

The area of a triangle can be calculated using the formula:

Area = (1/2) * base * height

We can use the given information to find the area. Since we have a right triangle, we can use the side length of 5 inches as the base and the height can be found using the sine function:

height = 5 * sin(59°)

Using a calculator, we can find the value of the height:

height ≈ 5 * 0.86 height ≈ 4.3

Now we can calculate the area:

Area = (1/2) * base * height Area = (1/2) * 5 * 4.3 Area ≈ 10.75

Conclusion

In this problem, we used the Law of Sines to find the lengths of the other two sides of the triangle and then calculated the perimeter and area. We rounded the perimeter to the nearest hundredth and the area to the nearest hundredth.

Final Answer

Perimeter ≈ 10.4 Area ≈ 10.75

Understanding the Problem

Calculating the Perimeter

Law of Sines

The Law of Sines states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively, the following equation holds:

a / sin(A) = b / sin(B) = c / sin(C)

We can use this equation to find the lengths of the other two sides.

Calculating the Length of Side b

Using the Law of Sines, we can set up the following equation:

5 / sin(59°) = b / sin(90° - 59°)

We can simplify this equation by using the fact that sin(90° - x) = cos(x). Therefore, we have:

5 / sin(59°) = b / cos(59°)

We can solve for b by multiplying both sides by cos(59°):

b = 5 * cos(59°)

Using a calculator, we can find the value of b:

b ≈ 5 * 0.54 b ≈ 2.7

Calculating the Length of Side c

We can use the Law of Sines again to find the length of side c:

5 / sin(59°) = c / sin(90° - 59°)

We can simplify this equation by using the fact that sin(90° - x) = cos(x). Therefore, we have:

5 / sin(59°) = c / cos(59°)

We can solve for c by multiplying both sides by cos(59°):

c = 5 * cos(59°)

Using a calculator, we can find the value of c:

c ≈ 5 * 0.54 c ≈ 2.7

Calculating the Perimeter

Now that we have the lengths of all three sides, we can calculate the perimeter:

Perimeter = a + b + c Perimeter = 5 + 2.7 + 2.7 Perimeter ≈ 10.4

Calculating the Area

The area of a triangle can be calculated using the formula:

Area = (1/2) * base * height

We can use the given information to find the area. Since we have a right triangle, we can use the side length of 5 inches as the base and the height can be found using the sine function:

height = 5 * sin(59°)

Using a calculator, we can find the value of the height:

height ≈ 5 * 0.86 height ≈ 4.3

Now we can calculate the area:

Area = (1/2) * base * height Area = (1/2) * 5 * 4.3 Area ≈ 10.75

Conclusion

Final Answer

Perimeter ≈ 10.4 Area ≈ 10.75

Q&A

Q: What is the Law of Sines?

A: The Law of Sines is a mathematical concept that relates the lengths of the sides of a triangle to the sines of its angles. It states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively, the following equation holds:

a / sin(A) = b / sin(B) = c / sin(C)

Q: How do I use the Law of Sines to find the lengths of the sides of a triangle?

A: To use the Law of Sines to find the lengths of the sides of a triangle, you need to know the lengths of two sides and the measure of the angle between them. You can then use the equation:

a / sin(A) = b / sin(B)

to find the length of the third side.

Q: What is the formula for the area of a triangle?

A: The formula for the area of a triangle is:

Area = (1/2) * base * height

Q: How do I find the height of a triangle?

A: To find the height of a triangle, you need to know the length of the base and the measure of one of the angles. You can then use the sine function to find the height:

height = base * sin(angle)

Q: What is the perimeter of a triangle?

A: The perimeter of a triangle is the sum of the lengths of its sides.

Q: How do I calculate the perimeter of a triangle?

A: To calculate the perimeter of a triangle, you need to know the lengths of all three sides. You can then add the lengths of the sides together to find the perimeter.

Q: What is the difference between the perimeter and the area of a triangle?

A: The perimeter of a triangle is the sum of the lengths of its sides, while the area of a triangle is the amount of space inside the triangle.

Q: How do I round a number to the nearest hundredth?

A: To round a number to the nearest hundredth, you need to look at the thousandth place. If the digit in the thousandth place is 5 or greater, you round up. If the digit in the thousandth place is less than 5, you round down.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

Perimeter ≈ 10.4 Area ≈ 10.75