In Fig. 2.33, O Is The Centre Of Circle MLY. If OLY = 50° And OMY MOL. = 15°, Calculate MOL​

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Introduction

In geometry, circles and angles play a crucial role in understanding various mathematical concepts. Fig. 2.33 presents a scenario where O is the centre of circle MLY, and we are given two angles: OLY = 50° and OMY MOL. = 15°. Our objective is to calculate the value of MOL.

Understanding the Problem

To solve this problem, we need to apply the properties of circles and angles. The given information includes:

  • O is the centre of circle MLY.
  • OLY = 50°
  • OMY MOL. = 15°

We are required to find the value of MOL.

Applying Geometric Principles

To calculate MOL, we can use the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circle's circumference. This principle is known as the "angle at the centre is twice the angle at the circumference" theorem.

Step 1: Finding the Angle at the Centre

Let's denote the angle at the centre as ∠OLY. We know that ∠OLY = 50°.

Step 2: Finding the Angle at the Circumference

Since OMY MOL. = 15°, we can infer that the angle at the circumference, ∠OML, is half of the angle at the centre. Therefore, ∠OML = 15°.

Step 3: Calculating MOL

Now, we can use the fact that the sum of the angles in a triangle is 180° to find MOL. Let's denote the angle at M as ∠MOL. We know that ∠OML = 15° and ∠OLY = 50°.

Using the angle sum property, we can write:

∠MOL + ∠OML + ∠OLY = 180°

Substituting the known values, we get:

∠MOL + 15° + 50° = 180°

Simplifying the equation, we get:

∠MOL = 115°

Therefore, MOL = 115°.

Conclusion

In this problem, we applied geometric principles to calculate the value of MOL. By using the angle at the centre is twice the angle at the circumference theorem and the angle sum property, we were able to find the value of MOL.

Key Takeaways

  • The angle at the centre of a circle is twice the angle at the circumference.
  • The sum of the angles in a triangle is 180°.
  • By applying these principles, we can solve problems involving circles and angles.

Practice Problems

  1. In Fig. 2.34, O is the centre of circle MLY. If OLY = 30° and OMY MOL. = 20°, calculate MOL.
  2. In Fig. 2.35, O is the centre of circle MLY. If OLY = 40° and OMY MOL. = 25°, calculate MOL.

Solutions

  1. Using the same principles as above, we can calculate MOL as follows:

∠MOL + ∠OML + ∠OLY = 180°

Substituting the known values, we get:

∠MOL + 20° + 30° = 180°

Simplifying the equation, we get:

∠MOL = 130°

Therefore, MOL = 130°.

  1. Using the same principles as above, we can calculate MOL as follows:

∠MOL + ∠OML + ∠OLY = 180°

Substituting the known values, we get:

∠MOL + 25° + 40° = 180°

Simplifying the equation, we get:

∠MOL = 115°

Therefore, MOL = 115°.

Conclusion

Frequently Asked Questions

In this article, we will address some of the most common questions related to calculating MOL in Fig. 2.33.

Q: What is the angle at the centre of a circle?

A: The angle at the centre of a circle is twice the angle at the circumference. This is known as the "angle at the centre is twice the angle at the circumference" theorem.

Q: How do I calculate MOL using the angle at the centre is twice the angle at the circumference theorem?

A: To calculate MOL, you need to follow these steps:

  1. Find the angle at the centre, ∠OLY.
  2. Find the angle at the circumference, ∠OML.
  3. Use the angle sum property to find MOL.

Q: What is the angle sum property?

A: The angle sum property states that the sum of the angles in a triangle is 180°.

Q: How do I use the angle sum property to find MOL?

A: To use the angle sum property to find MOL, you need to follow these steps:

  1. Write the equation: ∠MOL + ∠OML + ∠OLY = 180°
  2. Substitute the known values: ∠MOL + 15° + 50° = 180°
  3. Simplify the equation: ∠MOL = 115°

Q: What if I have a different angle at the centre and circumference? How do I calculate MOL?

A: To calculate MOL with a different angle at the centre and circumference, you need to follow the same steps as above:

  1. Find the angle at the centre, ∠OLY.
  2. Find the angle at the circumference, ∠OML.
  3. Use the angle sum property to find MOL.

Q: Can I use the angle at the centre is twice the angle at the circumference theorem to find MOL in any triangle?

A: Yes, you can use the angle at the centre is twice the angle at the circumference theorem to find MOL in any triangle, as long as the triangle is inscribed in a circle.

Q: What if I have a triangle that is not inscribed in a circle? Can I still use the angle at the centre is twice the angle at the circumference theorem to find MOL?

A: No, you cannot use the angle at the centre is twice the angle at the circumference theorem to find MOL in a triangle that is not inscribed in a circle.

Q: Are there any other theorems or properties that I can use to find MOL?

A: Yes, there are other theorems and properties that you can use to find MOL, such as the exterior angle theorem and the interior angle theorem.

Conclusion

In this article, we addressed some of the most common questions related to calculating MOL in Fig. 2.33. We provided step-by-step instructions on how to use the angle at the centre is twice the angle at the circumference theorem and the angle sum property to find MOL. We also discussed other theorems and properties that you can use to find MOL.

Practice Problems

  1. In Fig. 2.36, O is the centre of circle MLY. If OLY = 25° and OMY MOL. = 20°, calculate MOL.
  2. In Fig. 2.37, O is the centre of circle MLY. If OLY = 35° and OMY MOL. = 25°, calculate MOL.

Solutions

  1. Using the same principles as above, we can calculate MOL as follows:

∠MOL + ∠OML + ∠OLY = 180°

Substituting the known values, we get:

∠MOL + 20° + 25° = 180°

Simplifying the equation, we get:

∠MOL = 135°

Therefore, MOL = 135°.

  1. Using the same principles as above, we can calculate MOL as follows:

∠MOL + ∠OML + ∠OLY = 180°

Substituting the known values, we get:

∠MOL + 25° + 35° = 180°

Simplifying the equation, we get:

∠MOL = 120°

Therefore, MOL = 120°.

Conclusion

In this article, we provided step-by-step instructions on how to use the angle at the centre is twice the angle at the circumference theorem and the angle sum property to find MOL. We also discussed other theorems and properties that you can use to find MOL. We provided practice problems and solutions to help readers understand and apply these principles.