Lesson 4-7: Medians, Altitudes, and Perpendicular Bisectors (page 152) Essential Questions Can you construct a proof using congruent triangles?

MEDIAN (of a triangle): a segment from a vertex to the midpoint of the opposite side. C B A

MEDIAN (of a triangle): a segment from a vertex to the midpoint of the opposite side. C B A

MEDIAN (of a triangle): a segment from a vertex to the midpoint of the opposite side. C B A

The name given to the point of intersection for the three medians is _____________________________________. a bonus question on the TEST C B A

Z Y X The three medians for an obtuse triangle.

ALTITUDE (of a triangle): the perpendicular segment from a vertex to the line containing the opposite side. C B A

C B A

C B A

The name given to the point of intersection for the three altitudes is _____________________________________. a bonus question on the TEST C B A

Z Y X The three altitudes for an obtuse triangle. YES … they do intersect!

F E The three altitudes for a right triangle. D

PERPENDICULAR BISECTOR (of a segment): a line, ray, or segment that is perpendicular to the segment at its midpoint. C B A

C B A

C B A

The name given to the point of intersection for the three ⊥ -Bisectors is _____________________________________. a bonus question on the TEST C B A

Z Y X The three ⊥ -Bisectors for an obtuse triangle.

BISECTOR of an ANGLE: the ray that divides the angle into two congruent adjacent angles. C B A

BISECTOR of an ANGLE: the ray that divides the angle into two congruent adjacent angles. C B A

C B A BISECTOR of an ANGLE: the ray that divides the angle into two congruent adjacent angles.

The name given to the point of intersection for the three ∠ -Bisectors is _____________________________________. a bonus question on the TEST C B A

Z Y X The three ∠ -Bisectors for an obtuse triangle.

If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Theorem 4-5 Given:l is the ⊥ -bisector of BC A is on l Prove: AB = AC A BC X l

Given:l is the ⊥ -bisector of BC A is on l Prove: AB = AC A BC X l Proof:To prove this theorem, the following triangles must be proven congruent … ∆ ABX ≅ ∆ ACX, by SAS Postulate, then AB = AC by CPCTC.

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Theorem 4-6 A BC

1 st way: A BC Proof:To prove this theorem, the following triangles must be proven congruent. X

A BC X 12 Given: AB = AC Prove:

Theorem 4-6 is the converse of Theorem 4-5 and can be combined into a biconditional. A point is on the perpendicular bisector of a segment ______________ it is equidistant from the endpoints of the segment. if and only if

DISTANCE from a POINT to a LINE (or a plane): the length of the perpendicular segment from the point to the line (or plane). t R NO!

If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 4-7 Prove: PX = PY A B C X Z P Y

StatementsReasons 1._______________________________________________ Given ________________________ 2._________________________________________________________________________________ 3._________________________________________________________________________________ 4._________________________________________________________________________________ 5._________________________________________________________________________________ 6.____________________________________ _____________________________________________ A B C X Z P Y ∆ BXP  ∆ BYP AAS Theorem Reflexive Property CPCTC Def. of Angle Bisector ∠ BXP  ∠ BYP Def. of ⊥ -lines & ≅ ∠ ’ s ∠ PBX  ∠ PBY Prove: PX = PY Proof:

If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. Theorem 4-8 A B C X P Y

A B C X P Y

Theorem 4-8 is the converse of Theorem 4-7 and can be combined into a biconditional. A point is on the bisector of an angle _______________ it is equidistant from the sides of the angle. if and only if

Assignment Written Exercises on pages 156 & 157 RECOMMENDED: 1 to 12 ALL numbers REQUIRED: 8, 9, 10, 12, 13, 19, 23 Prepare for Test on Chapter 4: Congruent Triangles Prepare for Quiz on Lessons 4-6 and 4-7 Can you construct a proof using congruent triangles?