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.

Definition: Power of a Point

Given a circle of radius π‘Ÿ centered at 𝑀 and a point 𝐴, the power of point 𝐴 with respect to circle 𝑀, denoted by 𝑃(𝐴), is given by 𝑃(𝐴)=π΄π‘€βˆ’π‘Ÿ.

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

Example 1: Finding the Power of a Point with Respect to a Circle

A circle has center 𝑀 and radius π‘Ÿ=21. Find the power of the point 𝐴 with respect to the circle given that 𝐴𝑀=25.

Answer

Recall that the power of a point 𝐴 with respect to a circle of radius π‘Ÿ with center 𝑀 is given by 𝑃(𝐴)=π΄π‘€βˆ’π‘Ÿ.

We are given that 𝐴𝑀=25 and π‘Ÿ=21. Substituting these values, we obtain 𝑃(𝐴)=25βˆ’21=184.

Hence, the power of point 𝐴 with respect to the circle 𝑀 is 184.

In the previous example, we computed the power of a point from the given lengths. We can see that the given length 𝐴𝑀 is greater than the radius of the circle, π‘Ÿ, which means that point 𝐴 is outside circle 𝑀. In this case, since 𝐴𝑀>π‘Ÿ, we also know that 𝐴𝑀>π‘ŸοŠ¨οŠ¨. Then, the power of point 𝑃(𝐴)=π΄π‘€βˆ’π‘Ÿ>0.

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, 𝐴𝑀>π‘Ÿ, so the power of the point will be positive in this case. If the point is on the circle, then 𝐴𝑀=π‘Ÿ, so the power of the point will be equal to zero. In the last case, if the point is inside the circle, then 𝐴𝑀<π‘Ÿ, so the power of the point will be negative.

Property: Sign of the Power of a Point

Consider a circle 𝑀 and a point 𝐴. The power of point 𝐴 with respect to circle 𝑀 is denoted by 𝑃(𝐴).

  • If 𝑃(𝐴)>0, then point 𝐴 lies outside circle 𝑀.
  • If 𝑃(𝐴)=0, then point 𝐴 lies on circle 𝑀.
  • If 𝑃(𝐴)<0, then point 𝐴 lies inside circle 𝑀.

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 𝐴 with respect to circle 𝑁 if 𝑃(𝐴)=814.

Answer

The power of point 𝐴 with respect to the circle with radius π‘Ÿ and center 𝑁 is given by 𝑃(𝐴)=π΄π‘βˆ’π‘Ÿ.

We are given that 𝑃(𝐴)=814. Particularly, this means that the power of point 𝐴 is positive. This tells us that π΄π‘βˆ’π‘Ÿ>0,𝐴𝑁>π‘Ÿ.whichimpliesthat

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

The position of point 𝐴 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.

Example 3: Finding the Radius of a Circle given the Power of a Point with Respect to It and the Distance between Its Center and That Point

A point is at a distance 40 from the center of a circle. If its power with respect to the circle is 81, what is the radius of the circle, rounded to the nearest integer?

Answer

The power of point 𝐴 with respect to the circle with radius π‘Ÿ and center 𝑀 is given by 𝑃(𝐴)=π΄π‘€βˆ’π‘Ÿ.

We are given that 𝐴𝑀=40 and 𝑃(𝐴)=81. Substituting these values into the equation above, we obtain 81=40βˆ’π‘Ÿ.

Rearranging this equation, we obtain π‘Ÿ=40βˆ’81=1519.

Taking the positive square root, since π‘Ÿ is a length, leads to π‘Ÿ=38.9744….

Hence, the radius of the circle rounded to the nearest integer is 39 length units.

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.

Property: Power of a Point and the Length of a Tangent Segment

Consider a circle 𝑀 and a point 𝐴 outside the circle. Let 𝐴𝐡 be a tangent segment to the circle at point 𝐡.

Then, 𝑃(𝐴)=𝐴𝐡.

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 𝐴𝐡𝑀 in the diagram is a right triangle and the Pythagorean theorem tells us that 𝐴𝐡+π‘Ÿ=𝐴𝑀,𝐴𝐡=π΄π‘€βˆ’π‘Ÿ.whichcanberearrangedas

The right-hand side of this equation is the power of point 𝑃(𝐴). 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.

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

Consider a circle 𝑀 and a point 𝐴 outside the circle. Let 𝐴𝐷 be a secant segment to the circle at points 𝐢 and 𝐷 respectively. Then, 𝑃(𝐴)=𝐴𝐢×𝐴𝐷.

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

We have also added chords 𝐡𝐢 and 𝐡𝐷 in blue to the diagram above. We claim that triangle 𝐴𝐡𝐢 is similar to triangle 𝐴𝐷𝐡. Since the angle at vertex 𝐴 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 𝐡𝐢 is denoted by πœƒ, the inscribed angle π‘šβˆ π΅π·πΆ=12πœƒ.

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 𝐴𝐡 and the chord 𝐢𝐡 is one half the measure of the arc 𝐡𝐢, which is πœƒ. This gives us π‘šβˆ π΄π΅πΆ=12πœƒ.

This tells us that the angles ∠𝐴𝐡𝐢 and ∠𝐡𝐷𝐢 are congruent, which proves the similarity β–³π΄π΅πΆβˆΌβ–³π΄π·π΅.

Using the similarity of these triangles, we can write an equation involving the ratios 𝐴𝐡𝐴𝐷=𝐴𝐢𝐴𝐡,𝐴𝐡=𝐴𝐢×𝐴𝐷.whichsimplifiesto

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.

Theorem: The Power of a Point Theorem for Tangent and Secant Segments

Consider a circle 𝑀 and a point 𝐴 outside the circle. Let 𝐴𝐡 be a tangent segment to a circle at point 𝐡 and 𝐴𝐷 be a secant segment to the circle at points 𝐢 and 𝐷 repsectively. Then, 𝐴𝐡=𝐴𝐢×𝐴𝐷.

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 𝐴𝐢 and 𝐴𝐷, which are parts of secant ⃖⃗𝐢𝐷 to the circle at 𝐢 and 𝐷, passing through the exterior point 𝐴. 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 𝐴𝐡 and a secant 𝐴𝐷 that cut the circle at 𝐢. Given that 𝐴𝐡=7cm and 𝐴𝐢=5cm, find the length of 𝐢𝐷. 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 𝐴 be a point outside the circle, and let 𝐡, 𝐢, and 𝐷 be points on the circle such that 𝐴𝐡 is a tangent segment and 𝐴𝐷 is a secant segment to the circle. Then, 𝐴𝐡=𝐴𝐢×𝐴𝐷.

We are given that 𝐴𝐡=7cm and 𝐴𝐢=5cm. Substituting these values into the equation above gives us 7=5×𝐴𝐷,𝐴𝐷=75=9.8.whichleadstocm

We want to find the length of 𝐢𝐷, and we can see that 𝐴𝐷=𝐴𝐢+𝐢𝐷. Substituting 𝐴𝐷=9.8cm and 𝐴𝐢=5cm into this equation gives us 9.8=5+𝐢𝐷,𝐢𝐷=9.8βˆ’5=4.8.whichleadstocm

Hence, the length of 𝐢𝐷 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.

Theorem: The Power of a Point Theorem for Two Secant Segments

Consider a circle 𝑀 and a point 𝐴 outside the circle. Let 𝐴𝐢 be a secant segment to the circle at 𝐡 and 𝐢, respectively, and 𝐴𝐸 be a secant segment to the same circle at 𝐷 and 𝐸 respectively. Then, 𝐴𝐡×𝐴𝐢=𝐴𝐷×𝐴𝐸.

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

By the property of the power of point 𝐴 on the secant segment 𝐴𝐢, we know that 𝑃(𝐴)=𝐴𝐡×𝐴𝐢.

If we apply the same property for the secant 𝐴𝐸, then we see that 𝑃(𝐴)=𝐴𝐷×𝐴𝐸.

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, 𝐴𝐡 and 𝐴𝐷, intersecting at 𝐴. Given that 𝐴𝐸=3cm, 𝐸𝐷=5cm, and 𝐴𝐡=9cm, find the length of 𝐡𝐢, 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 𝐴 be a point outside the circle. If 𝐡, 𝐢, 𝐷, and 𝐸 are points on the circle such that 𝐴𝐡 is a secant segment to the circle at 𝐢 and 𝐡, respectively, and 𝐴𝐷 is a secant segment to the same circle at 𝐸 and 𝐷, respectively, then 𝐴𝐢×𝐴𝐡=𝐴𝐸×𝐴𝐷.

We are given the lengths of the line segments 𝐴𝐸=3,𝐸𝐷=5,𝐴𝐡=9.cmcmandcm

Since 𝐴𝐸+𝐸𝐷=𝐴𝐷, we can also obtain 𝐴𝐷=3+5=8.cm

We are looking for the length of 𝐡𝐢, which is part of 𝐴𝐡. Since we know the length of 𝐴𝐡 and 𝐴𝐡=𝐴𝐢+𝐡𝐢, we can find the length of 𝐡𝐢 by first finding the length of 𝐴𝐢. If we substitute the values of 𝐴𝐡,𝐴𝐸, and 𝐴𝐷 into the equation from the power of a point theorem, we can find the length of 𝐴𝐢 as follows: 𝐴𝐢×9=3Γ—8,𝐴𝐢=3Γ—89=83.whichleadstocm

Substituting this value into the equation 𝐴𝐡=𝐴𝐢+𝐡𝐢, we obtain 9=83+𝐡𝐢,𝐡𝐢=9βˆ’83=27βˆ’83=193.whichleadstocm

Hence, the length of 𝐡𝐢 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 𝑀 and a point 𝐴 inside the circle. Let 𝐡𝐢 be a chord in the circle passing through point 𝐴 as shown below.

Then, βˆ’π‘ƒ(𝐴)=𝐴𝐡×𝐴𝐢.

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

The green chords 𝐡𝐷 and 𝐢𝐸 were added to the diagram. We claim that β–³π΄π΅π·βˆΌβ–³π΄πΈπΆ.

Since angles ∠𝐷𝐴𝐡 and ∠𝐸𝐴𝐢 are opposite angles, we know that they are congruent. Also, angles ∠𝐢𝐡𝐷 and ∠𝐷𝐸𝐢 are inscribed angles to the same arc 𝐢𝐷. 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 𝐴𝐡𝐴𝐸=𝐴𝐷𝐴𝐢, which leads to

𝐴𝐡×𝐴𝐢=𝐴𝐷×𝐴𝐸.(1)

Let us examine how this identity relates to the concept of the power of a point. Recall that the power of point 𝐴 with respect to the circle 𝑀 is given by 𝑃(𝐴)=π΄π‘€βˆ’π‘Ÿ.

We know that this value will be negative since point 𝐴 is inside circle 𝑀. 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: βˆ’π‘ƒ(𝐴)=π‘Ÿβˆ’π΄π‘€.

We can use the difference of the square formula, π‘Žβˆ’π‘=(π‘Žβˆ’π‘)(π‘Ž+𝑏), to simplify this expression to βˆ’π‘ƒ(𝐴)=(π‘Ÿβˆ’π΄π‘€)(π‘Ÿ+𝐴𝑀).

Looking at the diagram, we can see that 𝐷𝑀=π‘Ÿ, and subtracting 𝐴𝑀 from 𝐷𝑀 gives the length 𝐷𝐴. Similarly, we can see that 𝑀𝐸=π‘Ÿ, and adding 𝐴𝑀 to 𝑀𝐸 gives the length 𝐴𝐸. In other words, π‘Ÿβˆ’π΄π‘€=𝐷𝐴,π‘Ÿ+𝐴𝑀=𝐴𝐸.and

Substituting these expressions in the equation for βˆ’π‘ƒ(𝐴) above, we have βˆ’π‘ƒ(𝐴)=𝐷𝐴×𝐴𝐸.

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.

Theorem: The Power of a Point Theorem for Two Chords

Consider a circle 𝑀 and a point 𝐴 inside the circle. Let 𝐡𝐢 and 𝐷𝐸 be two intersecting chords at point 𝐴.

Then, 𝐴𝐡×𝐴𝐢=𝐴𝐷×𝐴𝐸.

To prove this theorem, consider the following diagram.

If we apply the property of the power of a point for point 𝐴 on chord 𝐡𝐢, we have βˆ’π‘ƒ(𝐴)=𝐴𝐡×𝐴𝐢.

If we apply the same property for chord 𝐷𝐸, then βˆ’π‘ƒ(𝐴)=𝐴𝐷×𝐴𝐸.

Since both quantities on the right-hand side of each equation are equal to βˆ’π‘ƒ(𝐴), 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, 𝐴𝐢 and 𝐡𝐷, intersecting at 𝐸. Given that 𝐴𝐸∢𝐡𝐸=1∢3 and 𝐢𝐸=6cm, find the length of 𝐷𝐸.

Answer

Recall the power of a point theorem that relates the lengths of line segments in two different chords: Let 𝐸 be a point inside the circle. If 𝐴, 𝐡, 𝐢, and 𝐷 are points on the circle such that 𝐴𝐢 and 𝐡𝐷 are chords to the circle, then 𝐴𝐸×𝐢𝐸=𝐷𝐸×𝐡𝐸.

We are given the length of 𝐢𝐸, but we are only given the ratio regarding the lengths of 𝐴𝐸 and 𝐡𝐸. Let us denote the length of 𝐴𝐸 by π‘₯ cm. Since 𝐴𝐸∢𝐡𝐸=1∢3, the length of 𝐡𝐸 must be equal to 3π‘₯ cm. Then, we have 𝐢𝐸=6,𝐴𝐸=π‘₯,𝐡𝐸=3π‘₯.cmcmandcm

Substituting these expressions into the equation from the power of a point theorem, we obtain π‘₯Γ—6=𝐷𝐸×(3π‘₯).

Since π‘₯β‰ 0, we can divide both sides of this equation by π‘₯, which leads to 6=𝐷𝐸×3.

This gives us 𝐷𝐸=63=2.

Hence, the length of 𝐷𝐸 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.

Key Points

  • Given a circle of radius π‘Ÿ centered at 𝑀 and a point 𝐴, the power of point 𝐴 with respect to circle 𝑀, denoted by 𝑃(𝐴), is given by 𝑃(𝐴)=π΄π‘€βˆ’π‘Ÿ.
  • Consider a circle 𝑀 and a point 𝐴. The power of point 𝐴 with respect to circle 𝑀 is denoted by 𝑃(𝐴).
    • If 𝑃(𝐴)>0, then point 𝐴 lies outside circle 𝑀.
    • If 𝑃(𝐴)=0, then point 𝐴 lies on circle 𝑀.
    • If 𝑃(𝐴)<0, then point 𝐴 lies inside circle 𝑀.
  • The power of a point theorem is stated in the following three parts.
    • Tangent and secant segments: Consider a circle 𝑀 and a point 𝐴 outside the circle. Let 𝐴𝐡 be a tangent segment to a circle and 𝐴𝐷 be a secant segment to the circle at 𝐢 and 𝐷 respectively. Then, 𝐴𝐡=𝐴𝐢×𝐴𝐷.
    • Two secant segments: Consider a circle 𝑀 and a point 𝐴 outside the circle. Let 𝐴𝐢 be a secant segment to the circle at 𝐡 and 𝐢, respectively, and 𝐴𝐸 be a secant segment to the same circle at 𝐷 and 𝐸 respectively. Then, 𝐴𝐡×𝐴𝐢=𝐴𝐷×𝐴𝐸.
    • Intersecting chords: Consider a circle 𝑀 and a point 𝐴 inside the circle. Let 𝐡𝐢 and 𝐷𝐸 be two intersecting chords at point 𝐴 in the circle. Then, 𝐴𝐡×𝐴𝐢=𝐴𝐷×𝐴𝐸.

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