Isosceles Triangles in Circles

1. The Fundamental Principle: Two Radii Make an Isosceles Triangle

The logic here is beautifully simple and stems directly from the definition of a circle. You already know what an isosceles triangle is and what a radius is. Let's formally connect them.

As the article explains, the logic is unassailable:

1. A circle is the set of all points equidistant from a center.
2. A radius is a line segment from the center to any point on that circle.
3. Therefore, by definition, all radii of the same circle are equal in length.
4. A triangle with two equal sides is an isosceles triangle.
5. Consequently, a triangle formed by two radii and a chord is always isosceles.

The most important consequence of this, which you'll use for calculations, is that the two angles at the base of the triangle (the angles opposite the radii) are equal.

2. How to Spot and Use the Property

Identifying the equal sides is the first step. The next is to correctly identify the equal angles. In a complex diagram, it's easy to get mixed up. The following video offers a simple, foolproof method.

Now, let's see this in action. It's a pattern you should constantly be looking for whenever a diagram includes the center of the circle.

3. Combining Concepts: A Multi-Step Example

This property of isosceles triangles rarely appears in isolation in an exam. More often, it's the first step in a chain of reasoning. You use the isosceles triangle to find one angle, which then allows you to use another circle theorem.

Let's look at a worked example that perfectly illustrates this.

This example shows why today's topic is so vital. Without spotting the isosceles triangle, you wouldn't have enough information to use the main circle theorem.

4. Your Turn: Practice Problems

Now, it's time to apply this yourself. Let's work through a few problems. For each one, try to solve it first, then read the solution. Remember to state your reasons, just as you would in an exam.

Problem 1
In a circle with center O, points A and B are on the circumference, forming triangle OAB. If angle OAB = 35°, find the size of angle AOB.

• Solution:
  • OA and OB are both radii of the circle, so they have equal length.
  • Therefore, triangle OAB is an isosceles triangle.
  • The base angles of an isosceles triangle are equal, so $\angle OBA=\angle OAB=35^{\circ }$. (Reason: Base angles of isosceles triangle)
  • The angles in a triangle sum to 180°. Therefore, $\angle AOB=180^{\circ }-(35^{\circ }+35^{\circ })=180^{\circ }-70^{\circ }=110^{\circ }$. (Reason: Sum of angles in a triangle)

Problem 2
In a circle with center O, points C and D are on the circumference. The angle at the center, $\angle COD$, is 84°. Find the size of angle OCD.

• Solution:
  • OC and OD are radii, so triangle OCD is isosceles.
  • The base angles, $\angle OCD$ and $\angle ODC$, are equal. (Reason: Base angles of isosceles triangle)
  • The sum of angles in the triangle is 180°. The sum of the base angles is $180^{\circ }-84^{\circ }=96^{\circ }$.
  • Since the base angles are equal, $\angle OCD=96^{\circ }/2=48^{\circ }$.

Problem 3 (Multi-step)
In a circle with center O, points P, Q, and R are on the circumference. You are given that angle OPQ = 20°. Find the size of angle PRQ.

• Solution:
  • Step 1: Consider triangle OPQ. OP and OQ are radii, so the triangle is isosceles.
  • This means $\angle OQP=\angle OPQ=20^{\circ }$. (Reason: Base angles of isosceles triangle)
  • Step 2: Find the angle at the center, $\angle POQ$.
  • $\angle POQ=180^{\circ }-(20^{\circ }+20^{\circ })=140^{\circ }$. (Reason: Sum of angles in a triangle)
  • Step 3: Now find the angle at the circumference, $\angle PRQ$. This angle is subtended by the same arc (PQ) as the central angle $\angle POQ$.
  • $\angle PRQ=\frac{1}{2}\times \angle POQ=\frac{1}{2}\times 140^{\circ }=70^{\circ }$. (Reason: Angle at circumference is half the angle at the center)

This last problem is a prime example of how you will be expected to chain together different geometric rules in your exam.

Conclusion

Excellent work today. We've focused on a simple but essential building block of circle geometry. You'll find yourself returning to this principle again and again as the problems become more complex.

Here are the key takeaways:

• The Core Idea: A triangle formed by two radii and a chord is always an isosceles triangle.
• The Key Property: Because the triangle is isosceles, the two angles opposite the radii (the "base angles") are equal.
• The Strategy: When you see a diagram with the center of the circle marked, immediately look for radii. If two radii form a triangle with a chord, you know it's isosceles and can start calculating angles.
• The Application: This property is often the first step in a multi-step problem, unlocking the information needed to apply other circle theorems.

In our next lesson, we will move on to shapes formed by four points on the circumference: cyclic quadrilaterals. You will learn the key theorem that governs the relationship between their opposite angles.