Calculating the Length of Side ZM

Oct 13, 2025 by wbrum.org 77 views

The Figure Shown Angle Y is a Right Angle and a Y-Z Equal 4 Inches. Angle YZM is 105 Degrees

Understanding the Problem

The problem presented involves a geometric figure with specific angle measurements and side lengths. We are given that angle Y is a right angle, and the length of side Y-Z is 4 inches. Additionally, we know that angle YZM measures 105 degrees. Our goal is to determine the diameter of a plug gauge that will touch all three sides of this figure.

Visualizing the Figure

To better understand the problem, let's visualize the figure described. We have a right-angled triangle YZM, with angle Y being 90 degrees. The length of side Y-Z is given as 4 inches. We also know that angle YZM measures 105 degrees. This information allows us to determine the length of side ZM using trigonometric principles.

Calculating the Length of Side ZM

Using the Law of Sines, we can calculate the length of side ZM. The Law of Sines states that for any triangle with angles A, B, and C, and opposite sides a, b, and c, respectively, the following equation holds:

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

In our case, we have:

YZ / sin(Y) = ZM / sin(YZM)

We know that YZ = 4 inches, angle Y = 90 degrees, and angle YZM = 105 degrees. Plugging in these values, we get:

4 / sin(90) = ZM / sin(105)

Since sin(90) = 1, we can simplify the equation to:

4 = ZM / sin(105)

To find the length of side ZM, we need to calculate sin(105). Using a calculator, we get:

sin(105) ≈ 0.9659

Now, we can plug this value back into the equation:

4 = ZM / 0.9659

Multiplying both sides by 0.9659, we get:

ZM ≈ 4 / 0.9659 ZM ≈ 4.13 inches

Determining the Diameter of the Plug Gauge

Now that we have the length of side ZM, we can determine the diameter of the plug gauge that will touch all three sides of the figure. Since the plug gauge will touch the sides Y-Z and ZM, its diameter must be equal to the length of side ZM.

Therefore, the diameter of the plug gauge is approximately 4.13 inches.

Conclusion

In this problem, we used trigonometric principles to calculate the length of side ZM, which is necessary to determine the diameter of the plug gauge. By applying the Law of Sines and using a calculator to find the sine of 105 degrees, we were able to find the length of side ZM and subsequently the diameter of the plug gauge.

Key Takeaways

The Law of Sines can be used to calculate the length of a side in a triangle when the lengths of the other two sides and the angles opposite those sides are known.
Trigonometric principles can be used to solve problems involving right-angled triangles and other types of triangles.
Calculators can be used to find the values of trigonometric functions such as sine, cosine, and tangent.

Further Reading

For more information on trigonometry and the Law of Sines, see the following resources:

"Trigonometry" by Michael Corral
"Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs
"Mathematics for the Nonmathematician" by Morris Kline

Glossary

Right angle: An angle that measures 90 degrees.
Law of Sines: A mathematical formula that relates the lengths of the sides of a triangle to the sines of its angles.
Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of triangles.
Plug gauge: A tool used to measure the diameter of a hole or other opening.
Diameter: The distance across a circle or other curved shape, passing through its center.
Frequently Asked Questions (FAQs) About the Figure Shown Angle Y is a Right Angle and a Y-Z Equal 4 Inches. Angle YZM is 105 Degrees

Q: What is the significance of angle Y being a right angle in this problem?

A: Angle Y being a right angle is significant because it allows us to use trigonometric principles to solve the problem. Specifically, it enables us to use the Law of Sines to calculate the length of side ZM.

Q: Why is the length of side Y-Z given as 4 inches?

A: The length of side Y-Z is given as 4 inches because it is a known value that is used as a reference point in the problem. It is used to calculate the length of side ZM using the Law of Sines.

Q: What is the relationship between the length of side ZM and the diameter of the plug gauge?

A: The length of side ZM is equal to the diameter of the plug gauge. This is because the plug gauge will touch the sides Y-Z and ZM, and its diameter must be equal to the length of side ZM.

Q: How can I use the Law of Sines to solve problems involving right-angled triangles?

A: To use the Law of Sines to solve problems involving right-angled triangles, you can follow these steps:

Identify the known values in the problem, including the lengths of the sides and the angles.
Use the Law of Sines to set up an equation that relates the known values.
Solve the equation to find the unknown value.
Use the unknown value to solve the problem.

Q: What are some common applications of trigonometry in real-world problems?

A: Trigonometry has many applications in real-world problems, including:

Navigation: Trigonometry is used in navigation to calculate distances and directions.
Physics: Trigonometry is used in physics to calculate the motion of objects.
Engineering: Trigonometry is used in engineering to design and build structures.
Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Q: How can I practice solving problems involving trigonometry?

A: To practice solving problems involving trigonometry, you can try the following:

Use online resources, such as Khan Academy or Mathway, to practice solving trigonometry problems.
Work with a tutor or teacher to practice solving trigonometry problems.
Use a calculator to check your answers and get feedback on your work.
Practice solving trigonometry problems on your own, using a textbook or other resource.

Q: What are some common mistakes to avoid when solving problems involving trigonometry?

A: Some common mistakes to avoid when solving problems involving trigonometry include:

Not using the correct formula or equation.
Not plugging in the correct values.
Not checking your work or using a calculator to check your answers.
Not using the correct units or measurements.

Q: How can I use trigonometry to solve problems involving right-angled triangles?

A: To use trigonometry to solve problems involving right-angled triangles, you can follow these steps:

Identify the known values in the problem, including the lengths of the sides and the angles.
Use the Pythagorean theorem to calculate the length of the hypotenuse.
Use the trigonometric ratios to calculate the lengths of the other sides.
Use the calculated values to solve the problem.

Q: What are some real-world applications of the Law of Sines?

A: The Law of Sines has many real-world applications, including:

Navigation: The Law of Sines is used in navigation to calculate distances and directions.
Physics: The Law of Sines is used in physics to calculate the motion of objects.
Engineering: The Law of Sines is used in engineering to design and build structures.
Computer Science: The Law of Sines is used in computer science to create 3D graphics and animations.

Q: How can I use the Law of Sines to solve problems involving non-right-angled triangles?

A: To use the Law of Sines to solve problems involving non-right-angled triangles, you can follow these steps:

Identify the known values in the problem, including the lengths of the sides and the angles.
Use the Law of Sines to set up an equation that relates the known values.
Solve the equation to find the unknown value.
Use the unknown value to solve the problem.

Q: What are some common challenges when using the Law of Sines to solve problems?

A: Some common challenges when using the Law of Sines to solve problems include:

Not using the correct formula or equation.
Not plugging in the correct values.
Not checking your work or using a calculator to check your answers.
Not using the correct units or measurements.

Q: How can I use trigonometry to solve problems involving 3D geometry?

A: To use trigonometry to solve problems involving 3D geometry, you can follow these steps:

Identify the known values in the problem, including the lengths of the sides and the angles.
Use the trigonometric ratios to calculate the lengths of the other sides.
Use the calculated values to solve the problem.
Use the Law of Sines to calculate the lengths of the sides of the triangle.

Q: What are some real-world applications of trigonometry in 3D geometry?

A: Trigonometry has many real-world applications in 3D geometry, including:

Computer-aided design (CAD): Trigonometry is used in CAD to create 3D models and animations.
Video games: Trigonometry is used in video games to create 3D graphics and animations.
Architecture: Trigonometry is used in architecture to design and build structures.
Engineering: Trigonometry is used in engineering to design and build machines and mechanisms.