# Question Video: Finding a Missing Length Using Equal Chords Mathematics

Given that π΄π΅ = πΆπ·, ππΆ = 10 cm, and π·πΉ = 8 cm, find the length of line segment ππΈ.

02:14

### Video Transcript

Given that π΄π΅ is equal to πΆπ·, ππΆ is equal to 10 centimeters, and π·πΉ is equal to eight centimeters, find the length of line segment ππΈ.

In this question, we are trying to calculate the length of ππΈ, which is the distance from the chord π΄π΅ to the center of the circle π. We begin by recalling that two chords of equal lengths are equidistant from the center. And in this question, we are told that the two chords π΄π΅ and πΆπ· are equal in length. This means that the length ππΉ must be equal to ππΈ. The line segment ππΈ perpendicularly bisects the chord π΄π΅. Likewise, ππΉ is the perpendicular bisector of πΆπ·. Since we are told π·πΉ is equal to eight centimeters, πΆπΉ, π΄πΈ, and π΅πΈ are all also equal to eight centimeters.

Our next step is to consider the right triangle ππΉπΆ. Using the Pythagorean theorem, ππΉ squared plus πΆπΉ squared is equal to ππΆ squared. By subtracting πΆπΉ squared from both sides and substituting in the values of πΆπΉ and ππΆ, we have ππΉ squared is equal to 10 squared minus eight squared. This is equal to 36. Square rooting both sides of this equation and knowing that ππΉ must be a positive answer, we have ππΉ is equal to six. Since ππΉ is equal to six centimeters, ππΈ must also be equal to six centimeters. The perpendicular distance from the center of the chord π΄π΅ to the center of the circle π, which is the line segment ππΈ, is equal to six centimeters.