Question Video: Using the Power of a Point Theorem for Secant Segments to Find the Power of a Point | Nagwa Question Video: Using the Power of a Point Theorem for Secant Segments to Find the Power of a Point | Nagwa

# Question Video: Using the Power of a Point Theorem for Secant Segments to Find the Power of a Point Mathematics • First Year of Secondary School

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Two circles π and π intersect at points π΄ and π΅, and the point πΆ satisfies πΆ β line π΅π΄ and πΆ β line segment π΅π΄. π· and πΈ are the points where line segment πΆπΈ intersects the circle π and πΆπΉ is a tangent to π. Given that πΆπ· = 7 and π·πΈ = 12, find π_(π) (πΆ).

04:21

### Video Transcript

Two circles π and π intersect at points π΄ and π΅, and the point πΆ satisfies πΆ belongs on the line π΅π΄ and πΆ does not belong on the line segment π΅π΄. π· and πΈ are the points where the line segment πΆπΈ intercects the circle π, and the line πΆπΉ is a tangent to π. Given that πΆπ· equals seven and π·πΈ equals 12, find π sub π of πΆ.

Thereβs an awful lot of information given to us in the question, so letβs begin by drawing a diagram. We have two circles with centers π and π. We donβt know which is larger. And actually, it doesnβt really matter. These two circles intersect at the points π΄ and π΅. Now weβre told that the point πΆ is on the line π΅π΄, but it isnβt on the line segment π΅π΄. That means if we draw in the line π΅π΄, πΆ is somewhere on this line, but it isnβt between π΄ and π΅. So perhaps itβs here. Thereβs then a line segment πΆπΈ, which intersects the circle π at points π· and πΈ, so we can add this line segment to our diagram. There is then a line πΆπΉ which is a tangent to circle π. So, hereβs that final line. The last information weβre given is the lengths of two line segments. πΆπ· is seven units and π·πΈ is 12 units.

So, we have a diagram. And now letβs look at what weβre being asked to find. π sub π of πΆ means the power of point πΆ with respect to circle π. Itβs calculated using the formula πΆπ squared minus π squared, where π is the radius of the circle π. So itβs the distance between points πΆ and the center of the circle squared minus the radius squared. However, we donβt have any of this information, so weβre going to need a different approach. Letβs consider circle π first as we have more information about this circle. In circle π, we know the lengths of πΆπ· and π·πΈ, which are each segments of a secant to this circle. We can therefore recall the power of a point theorem concerning the lengths of secant segments. This states the following.

Consider a circle π and a point πΆ outside the circle. Let the line segment πΆπΈ be a secant segment to the circle at π· and πΈ. Then the power of point πΆ with respect to circle π is equal to πΆπ· multiplied by πΆπΈ. This is great because we know the length of πΆπ·. Itβs seven units. And the length of πΆπΈ is seven plus 12. Itβs 19 units. So we can work out the power of point πΆ with respect to circle π as seven multiplied by 19, which is 133.

This isnβt what we were asked to find though. We were asked to find the power of point πΆ with respect to circle π. Well, the line at πΆπ΅ or π΅πΆ is also a secant of the circle π, so it follows that the product of the lengths of the secant segments πΆπ΄ and πΆπ΅ will also be equal to 133. This is because the power of a point with respect to any given circle is always the same. So we have the equation πΆπ΄ multiplied by πΆπ΅ is 133. But this is useful because πΆπ΅ isnβt just a secant segment of circle π; itβs also a secant segment of circle π. It is, in fact, a common secant segment for the two circles.

So by the power of a point theorem concerning the length of secant segments for circle π, we have that π π of πΆ is equal to πΆπ΄ multiplied by πΆπ΅. And weβve just determined that this will be equal to 133. In other words, what weβve found then is that the power of point πΆ with respect to each circle found using their common secant is the same. We can conclude then that the power of point πΆ with respect to circle π is 133.

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