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What two figures are created by drawing a median? What do these have in common?
Base Angles Theorem
Definition of angle congruence
Definition of segment congruence
ASA Congruence Theorem
Definition of angle bisector
Definition of perpendicular bisector
Let's first state the theorem.
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If a point is equidistant from the endpoints of a segment, then it lies on the segment's perpendicular bisector. |
First we recognize that, by the Base Angles Theorem, ∠ A and ∠ B are congruent.
Next, let's add a median passing through C and the midpoint of AB. Since AD is a bisector of AB, we now need to show that the two meet at a right angle.
This new line creates △ ACD and △ BCD. By the ASA Congruence Theorem, we can claim that these triangles are congruent. Since corresponding parts of congruent triangles are congruent, ∠ CDA≅ △ CDB. These angles are also supplementary making them both right angles.
This means that CD is the perpendicular bisector of AB. Since we drew AD such that C was on it, we know that C lies on the perpendicular bisector of AD. Now we can identify the definitions and theorems that we used. Base Angles Theorem Definition of angle congruence Definition of segment congruence ASA Congruence Theorem Definition of angle bisector Definition of perpendicular bisector