IF L is the circumcentre of `Delta XYZ` and angle X is `40^@` then the value of `angle YZL` is

Options

A. `50^@`
B. `60^@`
C. `40^@`
D. `70^@`

Answer

To solve the problem, we need to find the value of angle YZL in triangle XYZ, where L is the circumcenter and angle X is given as 40 degrees.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have triangle XYZ.
- Angle X = 40 degrees.
- L is the circumcenter of triangle XYZ.

**Hint**: Remember that the circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect.

2. **Use the Property of the Circumcenter**:
- The circumcenter L is equidistant from all three vertices of the triangle (X, Y, Z). This means that LY = LZ (both are radii of the circumcircle).

**Hint**: The circumcenter creates two isosceles triangles (LYZ and LZX) because the sides from L to Y and L to Z are equal.

3. **Calculate Angle YLZ**:
- Since angle X is 40 degrees, angle YLZ (which is the angle subtended by side YZ at the circumcenter) is twice angle X due to the property of angles subtended by the same arc in a circle.
- Therefore, angle YLZ = 2 * angle X = 2 * 40 degrees = 80 degrees.

**Hint**: Remember that the angle at the circumcenter is twice the angle at the circumference subtended by the same arc.

4. **Use the Triangle Angle Sum Property**:
- In triangle LYZ, the sum of angles is 180 degrees.
- Let angle YZL = θ. Then we can write the equation:
\[ angle LYZ + angle LZY + angle YZL = 180 \]
- We know angle LYZ = angle LZY (since LY = LZ), and both are equal to (180 - θ - 80)/2.

**Hint**: The angles opposite equal sides of an isosceles triangle are equal.

5. **Set Up the Equation**:
- Since angle LYZ = angle LZY, we can denote both as (180 - θ - 80)/2. Thus:
\[ 80 + 2 * (180 - θ - 80)/2 = 180 \]

6. **Solve for θ**:
- Simplifying the equation:
\[ 80 + (180 - θ - 80) = 180 \]
\[ 180 - θ = 100 \]
\[ θ = 80 \]

7. **Find angle YZL**:
- Now, we can find angle YZL:
\[ angle YZL = 180 - angle LYZ - angle LZY = 180 - 80 = 100 degrees \]

**Hint**: Always check the calculations and ensure that the angles add up to 180 degrees.

### Final Answer:
The value of angle YZL is 100 degrees.