Angles properties

Angles Properties

Circle Theorems of Inscribed Angles

An inscribed angle is formed when two secant lines intersect on a circle. It can also be formed using a secant line and a tangent line intersecting on a circle. A central angle, on the other hand, is an angle whose vertex is at the center of the circle and whose sides pass through a pair of points on the circle, therefore subtending an arc.

We explore the relationship between inscribed angles and central angles that subtend the same arc. The angle of the subtended arc is the same as the measure of the central angle (by definition).

In the first circle, a central angle is subtended by arc. The angle is an inscribed angle subtended by the same arc. In the second circle, an inscribed angle and a central angle both subtend the same arc. These observations lead us to a key result:

What can you say about the two angles subtending the same arc? You can draw several cases of central angles and inscribed angles subtending the same arc and measure them. Use dynamic geometry software if necessary. Are your observations the same?

Theorem

The measure of an angle inscribed in a circle is half the measure of the arc it intercepts. Note that this is equivalent to the measure of the inscribed angle being half the measure of the central angle if they intercept the same arc.

Proof

Let an inscribed angle and be a central angle both subtending arc as shown in the figure. Draw line. This forms two isosceles triangles and since two of their sides are radii of the circle.

In triangle, if we let the measure of be, then angle is also. By the exterior angle theorem, the measure of angle. This is also similar to triangle. If we let angle, it follows that is equal to 2y. In effect, the measure of the inscribed angle and the measure of central angle which is what we want to prove.

The Circle Theorems in Solving Related Problems

Example 1

An arc subtends an angle of 200° at the center of the circle, and the radius of the circle is 25 cm. Find the length of this arc.

Solution

Given:

• Radius $r = 25$ cm
• Central angle $\theta = 200°$

The length of an arc is given by the formula:

Arc length = $r \times \theta \times (\pi/180)$

Substitute the known values:

Arc length = $25 \times 200 \times (\pi/180)$

Arc length = $25 \times 200 \times (3.14159/180)$

Arc length = $25 \times 200 \times 0.0174533$

Arc length $\approx 8.73$ cm

So, the length of the arc is approximately $8.73$ cm.

Example 2

An arc of length 5 cm subtends an angle of 50° at the center of the circle. What is the radius of the circle?

Solution

Given:

• Arc length $l = 5$ cm
• Central angle $\theta = 50°$
• Radius $r = ?$

The formula for the arc length is:

Arc length = $r \times \theta \times (\pi/180)$

Substitute the known values:

5 = $r \times 50 \times (\pi/180)$

5 = $r \times 50 \times 0.0174533$

5 = $r \times 0.872$

Now solve for $r$:

$r \approx 5 / 0.872$

$r \approx 5.73$ cm

So, the radius of the circle is approximately $5.73$ cm.