Proof: perpendicular radius bisects chord

TL;DR

Segment OD is proven to bisect segment AC in a circle with the help of congruent triangles.

Transcript

• [Instructor] So we have this circle called circle O, based on the point at its center, and we have the segment OD, and we're told that segment OD, is a radius of circle O, fair enough. We're also told that segment OD is perpendicular to this chord to chord AC, or two segment AC. And what we wanna prove is that segment OD bisects AC. So another wa... Read More

Key Insights

• 🟰 In a circle, the length of a radius does not change, so AO and OC (both radii) are equal.
• 👻 Reflexivity allows us to state that a side is congruent to itself, as in the case of OM in the given triangles.
• 🔺 While congruent sides alone are not enough to establish the congruence of triangles, in the case of right triangles, it is sufficient.
• 🙃 The Pythagorean theorem can also be used to determine the length of a side in a right triangle if two sides are known.
• 🔺 The RSH postulate states that if two right triangles have a pair of congruent sides and a congruent angle, then the two triangles are congruent.
• 🙃 Congruent triangles have corresponding sides that are congruent.

Questions & Answers

Q: How is segment OD related to circle O?

Segment OD is a radius of circle O, meaning it connects the center O of the circle to a point D on its circumference.

Q: What information is given about segment AC?

It is stated that segment OD is perpendicular to segment AC, meaning they form a right angle where they meet.

Q: How are congruent triangles used in the proof?

Two congruent triangles (AMO and CMO) are established using the properties of the circle and the given right angle. If two triangles are congruent, then their corresponding sides are also congruent.

Q: How is it shown that OD bisects AC?

Because of congruence, it is proven that segment AM is congruent to segment CM. Therefore, point M, where OD intersects AC, must be the midpoint of AC, meaning OD bisects AC.

Summary & Key Takeaways

• The video discusses a circle with point O as its center, segment OD as its radius, and segment AC as a chord.

• The goal is to prove that segment OD bisects segment AC, meaning it intersects AC at its midpoint.

• By establishing congruence between two triangles and using the properties of right triangles, it is shown that OD indeed bisects AC.