Lesson Explainer: The Power of a Point Theorem Mathematics

In this explainer, we will learn how to find the power of a point with respect to a circle.

In plane geometry, we often encounter problems dealing with lengths of line segments involving circles. Many of the tools used to solve these types of problems are related to the concept of the power of a point. The power of a point is a real number that quantifies a geometric relationship between a point and a circle. This number is defined using the radius of the circle and the distance of the point from the center of the circle. As we will see in this explainer, this number also relates to the lengths involving secants, tangents, and chords of a circle.

In our first example, we will compute the power of a point when we are given these lengths.

In the previous example, we computed the power of a point from the given lengths. We can see that the given length is greater than the radius of the circle, , which means that point is outside circle . In this case, since , we also know that . Then, the power of point

In general, this tells us that the power of a point is positive if the point lies outside of the circle. Let us draw a diagram involving the lengths involved in calculating the power of a point in three different cases.

If the point lies outside of the circle, , so the power of the point will be positive in this case. If the point is on the circle, then , so the power of the point will be equal to zero. In the last case, if the point is inside the circle, then , so the power of the point will be negative.

In our next example, we will determine the relative position of a point with respect to a circle when we are given the power of the point.

Example 2: Determining the Position of a Point with Respect to a Circle given Its Power

Determine the position of point with respect to circle if .

Answer

The power of point with respect to the circle with radius and center is given by

We are given that . Particularly, this means that the power of point is positive. This tells us that

Hence, the distance between point and the center of circle is greater than the radius of the circle. This means that point lies outside of the circle. For instance, we can visualize this fact using the following diagram.

The position of point is outside the circle.

In the next example, we will find the radius of a circle when we are given the power of a point and the distance between the point and the center of the circle.

So far, we have considered a few examples to become more familiar with the concept of the power of a point with respect to a circle. We now turn our focus to various applications of the power of a point. The first property relates the power of a point with the lengths of a tangent segment from the point to a circle.

Let us prove this property. We can begin by considering the following diagram that depicts the relationship between the power of a point and the length of a tangent segment.

We know that the radius of a circle intersects perpendicularly with any tangent. So, the triangle in the diagram is a right triangle and the Pythagorean theorem tells us that

The right-hand side of this equation is the power of point . This proves the property stated above.

Next property of the power of a point relates it with the length of a secant segment from the point through the circle.

Let us prove this property. Consider the diagram below where is a point outside a circle centered at , is a tangent segment to the circle at , and is a secant segment that intersects the circle at points and respectively.

We have also added chords and in blue to the diagram above. We claim that triangle is similar to triangle . Since the angle at vertex is shared by both triangles, we only need to show that one other pair of angles is congruent to prove the claimed similarity.

Recall that an angle inscribed by an arc has half the measure of the intercepted arc. So, if the measure of arc is denoted by , the inscribed angle

We also know that the measure of an angle of tangency between a tangent and a chord is half the measure of the arc intercepted by the chord. Hence, the angle between the tangent segment and the chord is one half the measure of the arc , which is . This gives us

This tells us that the angles and are congruent, which proves the similarity

Using the similarity of these triangles, we can write an equation involving the ratios

Noting that the left side of this equation is equal to the power of a point, we have proved the property as stated above.

When a point is outside a circle, then the power of the point establishes a relation between the lengths involving tangent and secant segments. This is known as the power of a point theorem. The power of a point theorem consists of three different statements, but they all relate to the concept of the power of a point. Let us first examine the power of a point theorem for a tangent and a secant to the circle.

To prove this theorem, recall that if a point is outside a circle, the power of the point is equal to the square of the length of a tangent segment. Additionally, we observed that the power of a point is also equal to the product of the lengths of and , which are parts of secant to the circle at and , passing through the exterior point . Since both quantities are equal to the power of the point, we can conclude that they are equal to each other, which leads to the statement of the theorem above.

Let us consider an example where we will use this statement to find a missing length involving a tangent and a secant to a circle.

Example 4: Using the Power of a Point Theorem for a Tangent and a Secant to Find Missing Lengths

A circle has a tangent and a secant that cut the circle at . Given that and , find the length of . Given your answer to the nearest hundredth.

Answer

Recall the power of a point theorem that relates the lengths of line segments in a tangent and a secant: Let be a point outside the circle, and let , , and be points on the circle such that is a tangent segment and is a secant segment to the circle. Then,

We are given that and . Substituting these values into the equation above gives us

We want to find the length of , and we can see that . Substituting and into this equation gives us

Hence, the length of rounded to the nearest hundredth is 4.80 cm.

In the previous example, we used the power of a point theorem that relates the lengths of tangent and secant segments to the circle. The next statement of the power of a point theorem deals with lengths of line segments from two different secants.

To prove this theorem, consider the following diagram containing two secants, and .

By the property of the power of point on the secant segment , we know that

If we apply the same property for the secant , then we see that

Since both quantities are equal to the power of the point, they must be equal to each other. This proves the statement of the theorem above.

Let us consider an example where we need to apply this version of the power of a point theorem to find a missing length in a diagram involving two secants.

Example 5: Using the Power of a Point Theorem for Two Secants to Find Missing Lengths

A circle has two secants, and , intersecting at . Given that , , and , find the length of , giving your answer to the nearest tenth.

Answer

Recall the power of a point theorem that relates the lengths of line segments in two different secants: Let be a point outside the circle. If , , , and are points on the circle such that is a secant segment to the circle at and , respectively, and is a secant segment to the same circle at and , respectively, then

We are given the lengths of the line segments

Since , we can also obtain

We are looking for the length of , which is part of . Since we know the length of and , we can find the length of by first finding the length of . If we substitute the values of , and into the equation from the power of a point theorem, we can find the length of as follows:

Substituting this value into the equation , we obtain

Hence, the length of rounded to the nearest tenth is 6.3 cm.

The previous two statements of the power of a point theorem used properties of the power of a point outside the circle. We now turn our attention to a property of the power of the point when the point lies inside the circle.

Property: Power of a Point and the Lengths of Chord Segments

Consider a circle and a point inside the circle. Let be a chord in the circle passing through point as shown below.

Then,

Let us prove this property. Consider the following diagram that contains chord intersecting with a diameter of the circle.

The green chords and were added to the diagram. We claim that

Since angles and are opposite angles, we know that they are congruent. Also, angles and are inscribed angles to the same arc . Since we know that all inscribed angles of an arc have the same measure, these angles are also congruent. This proves the claimed similarity.

Using the similar triangles, we can write the following equation of the deduced ratios from the correct similarity statement as which leads to

Let us examine how this identity relates to the concept of the power of a point. Recall that the power of point with respect to the circle is given by

We know that this value will be negative since point is inside circle . In order to discuss this quantity in terms of lengths of line segments that are positive, let us consider instead the negative of this number:

We can use the difference of the square formula, , to simplify this expression to

Looking at the diagram, we can see that , and subtracting from gives the length . Similarly, we can see that , and adding to gives the length . In other words,

Substituting these expressions in the equation for above, we have

We note that the right-hand side of this equation is equal to the right-hand side of equation (1). This tells us that the negative of the power of a point is equal to the left-hand side of equation (1). This proves the property stated above.

This property leads to our last statement of the power of a point theorem involving two intersecting chords.

To prove this theorem, consider the following diagram.

If we apply the property of the power of a point for point on chord , we have

If we apply the same property for chord , then

Since both quantities on the right-hand side of each equation are equal to , they must be equal to each other. This proves to the last statement of the power of a point theorem.

Let us consider an example where we need to apply this version of the power of a point theorem to find a missing length in a diagram involving two intersecting chords.

Example 6: Using the Power of a Point Theorem for Two Chords to Find Missing Lengths

A circle has two chords, and , intersecting at . Given that and , find the length of .

Answer

Recall the power of a point theorem that relates the lengths of line segments in two different chords: Let be a point inside the circle. If , , , and are points on the circle such that and are chords to the circle, then

We are given the length of , but we are only given the ratio regarding the lengths of and . Let us denote the length of by cm. Since , the length of must be equal to cm. Then, we have

Substituting these expressions into the equation from the power of a point theorem, we obtain

Since , we can divide both sides of this equation by , which leads to

This gives us .

Hence, the length of is 2 cm.

In this explainer, we have observed a variety of useful geometric properties based on the power of a point. Let us finish by recapping a few important concepts from this explainer.