Power of a Point

Links and useful resources

• gr10's Geometry >>>START HERE<<<
• AoPS Online Textbook
• AoPS Alcumus
• Big Ideas Geometry textbook
• GeoGebra Online Geometry Constuction Tool
• Two-column math templates
• Proofs unit slides from mathgiraffe.com
• Proofs unit printables for two-column proofs

Concept summary and connections

• Length of chords that meet in a circle
• Length of secants that cross outside of a circle
• The product of the two parts (each side of the point) of a chord through a point is the same or all chords through that point
• The square of a tangent through a point is equal to the product of the lengths of the parts of any secant through the same point (the part inside the circle times the part outside)

Lesson and worked examples

We've learned a lot of stuff about the properties of circles with angles, tangents, secants, and chords. We're going to put a lot of that together now into some useful rules for lengths of segments involved in all of those things.

Product of partial chord lengths through a point is constant

In the circle below, we will prove that for any chord through X, the product of the two parts of the chord is the same. In other words, $AX\times XB=DX\times XC$, no matter what the actual chords are!*

Tangent squared equals secant partial lengths product

In the diagram below, $(PA{)}^2=PB\times BC=PD\times DE$

Media resources

• Youtube search for "Length of chords that meet in a circle"
• Youtube search for "Length of secants that cross outside of a circle"
• Youtube search for "The product of the two parts (each side of the point) of a chord through a point is the same or all chords through that point"
• Youtube search for "The square of a tangent through a point is equal to the product of the lengths of the parts of any secant through the same point (the part inside the circle times the part outside)"

Guided practice

• Prove that the product of the two parts of a chord through X is the same for any chord through X
• Prove that if X = length of a tangent through point outside a circle, and Y,Z = the lengths of the inner and outer parts of a secant through the same point, then $X^2=Y\times Z$
• C is outside a circle. CA is a secant that passes through the circle at B, and CE is a secant that hits at D. CB=9, BA=11, CE=18. What is DE?
  

Homework

• #hw (geom-gr10) 13.1 What is Power of a Point? (p. 340) 2025-02-26