Exercise 24 Page 738

Let's consider a circle centered at P, a chord KM, and a radius JP such that KM⊥ JP.

Next, let's draw the radii KP and MP which are congruent. Also, LP≅LP by the Reflexive Property of Congruence.

Next, since △ LKP and △ LMP are right triangles, we can apply the Hypotenuse-Leg Congruence Theorem to conclude that these triangles are congruent. △ LKP ≅ △ LMP ↙ ↘ LK ≅ LM ∠ KPL ≅ ∠ MPL We can conclude that L is the midpoint of KM and that JP bisects KM. Also, by the definition of arc measure, we can write the following two equations. mKJ = m∠ KPL mJM = m∠ MPL Since the measures written on the right-hand side are equal, we conclude that both arcs have the same measure. This implies that JP bisects KM.

Two-Column Proof

Given: & ⊙ P, KM⊥JP Prove: & JP bisectsKMandKM Let's summarize the proof we did in the following two-column table.