Understanding the Geometry Problem: Circumcenter and Angles

This question asks us to find the measure of a specific angle within a triangle, given information about its circumcenter and another angle. We are given a triangle \(\Delta XYZ\), its circumcenter L, and a perpendicular LM drawn from L to side YZ. We are also given that \(\angle YXZ = 60^\circ\). We need to determine the measure of \(\angle YLM\).

Key Concepts for Solving

• Circumcenter: The circumcenter of a triangle is the intersection point of the perpendicular bisectors of its sides. It is also the center of the circle that passes through all three vertices of the triangle (the circumcircle).
• Angle at the Center vs. Angle at the Circumference: The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circle's circumference.
• Isosceles Triangle Properties: In an isosceles triangle, the angle bisector of the vertex angle, the median to the base, and the altitude to the base all coincide.

Step-by-Step Solution

Let's break down the problem step by step using the properties of the circumcenter and triangles.

Step 1: Relate the Circumcenter to the Vertices

L is the circumcenter of \(\Delta XYZ\). By definition, the circumcenter is equidistant from the vertices of the triangle. Therefore, LY = LZ = LX (all are radii of the circumcircle).

Step 2: Use the Angle Subtended by the Arc YZ

Consider the circumcircle passing through points X, Y, and Z with center L. The angle subtended by the arc YZ at the circumference is \(\angle YXZ = 60^\circ\). The angle subtended by the same arc YZ at the center L is \(\angle YLZ\).

According to the property relating the angle at the center and the angle at the circumference:

\[ \angle YLZ = 2 \times \angle YXZ \] \[ \angle YLZ = 2 \times 60^\circ \] \[ \angle YLZ = 120^\circ \]

Step 3: Analyze Triangle LYZ

Since LY = LZ (radii of the circumcircle), \(\Delta LYZ\) is an isosceles triangle with vertex angle \(\angle YLZ = 120^\circ\).

Step 4: Use the Perpendicular from the Circumcenter

We are given that LM is drawn perpendicular to YZ. So, \(\angle LMY = \angle LMZ = 90^\circ\).

In an isosceles triangle \(\Delta LYZ\), the altitude from the vertex L to the base YZ (which is LM) has special properties. It acts as the angle bisector of the vertex angle \(\angle YLZ\) and also as the median to the base YZ (meaning M is the midpoint of YZ).

Step 5: Find the Measure of Angle YLM

Since LM bisects the angle \(\angle YLZ\), we have:

\[ \angle YLM = \frac{1}{2} \times \angle YLZ \] \[ \angle YLM = \frac{1}{2} \times 120^\circ \] \[ \angle YLM = 60^\circ \]

Thus, the measure of \(\angle YLM\) is 60°.

Conclusion

By using the property that the angle subtended by an arc at the circumcenter is twice the angle subtended at the circumference, and the properties of an isosceles triangle formed by the circumcenter and two vertices, we found that \(\angle YLM\) is 60°.

Revision Table: Key Geometric Concepts

Additional Information: Circumcenter Location

The location of the circumcenter L depends on the type of triangle \(\Delta XYZ\):

• Acute Triangle: The circumcenter lies inside the triangle.
• Right Triangle: The circumcenter lies on the midpoint of the hypotenuse.
• Obtuse Triangle: The circumcenter lies outside the triangle.

In our problem, \(\angle YXZ = 60^\circ\), which is an acute angle. However, we don't know if the triangle is acute, right, or obtuse based on just one angle. The calculation of \(\angle YLZ\) and \(\angle YLM\) remains valid regardless of the triangle type, as the angle subtended by an arc at the center rule applies generally, and \(\Delta LYZ\) will always be isosceles with LM being the altitude from L to YZ.