Bisectors Concept 35 Perpendicular bisector a segment line

Bisectors Concept 35

• Perpendicular bisector – a segment, line or plane that intersects a segment at its midpoint and creates right angles.

Use the Perpendicular Bisector Theorems A. Find BC. BC = AC Perpendicular Bisector Theorem BC = 8. 5 Substitution Answer: 8. 5

Use the Perpendicular Bisector Theorems B. Find XY. Answer: 6

Use the Perpendicular Bisector Theorems C. Find PQ. PQ = RQ 3 x + 1 = 5 x – 3 Substitution 1 = 2 x – 3 Subtract 3 x from each side. 4 = 2 x Add 3 to each side. 2 =x Divide each side by 2. So, PQ = 3(2) + 1 = 7. Answer: 7 Perpendicular Bisector Theorem

A. Find NO. A. 4. 6 B. 9. 2 C. 18. 4 D. 36. 8

B. Find TU. A. 2 B. 4 C. 8 D. 16

C. Find EH. A. 8 B. 12 C. 16 D. 20

• Angle bisector – a line bisects the angle of a triangle and divides it into 2 equal angles.

Use the Angle Bisector Theorems A. Find DB. DB = DC Angle Bisector Theorem DB = 5 Substitution Answer: DB = 5

Use the Angle Bisector Theorems B. Find m WYZ XYW Definition of angle bisector m WYZ = m XYW Definition of congruent angles m WYZ = 28 Substitution

Use the Angle Bisector Theorems C. Find QS. QS = SR 4 x – 1 = 3 x + 2 x– 1 =2 x =3 Angle Bisector Theorem Substitution Subtract 3 x from each side. Add 1 to each side. Answer: So, QS = 4(3) – 1 or 11.

• Angle Bisectors of a triangle – Angle bisectors do have a vertex as an endpoint.

• Perpendicular bisector – a segment, line or plane that intersects a segment at its midpoint and creates right angles.

• Perpendicular Bisectors of Triangles – Perpendicular bisectors do NOT have a vertex as an endpoint (no endpoints) – Circumcenter is point where perpendicular bisector meet at one point. – Where is circumcenter located? • Acute Triangle – Inside triangle • Right Triangle – On the midpoint of the hypotenuse • Obtuse Triangle – Outside the longest side of the triangle

Perpendicular Bisector Theorem DA = BD = CD

Example 1 The perpendicular bisectors of Find PX

Example 2 The perpendicular bisectors of meet at point G. Find GA.

Use the Circumcenter Theorem GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points.

Use the Circumcenter Theorem Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle. C Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.

• Angle Bisectors of a triangle – Angle bisectors do have a vertex as an endpoint. – The Incenter is the point at which angle bisector of a triangle meet at one point. – Where is the incenter located? • Always inside the triangle for any type of triangle.

Angle Bisector Theorem

Example 1 The angle bisector of ABC meet at point G. m DAG = ______ FG = ______ m ECG = ______ GD = ______ AF = ______ m GBF = _____

Example 2 The angle bisectors of XYZ meet at point P. Find PM. Find PY.

Use the Incenter Theorem Find m SPU if S is the incenter of ΔMNP. Find SR

A. Find the measure of GF if D is the incenter of ΔACF. A. 12 B. 144 C. 8 D. 65

B. Find the measure of BCD if D is the incenter of ΔACF. A. 58° B. 116° C. 52° D. 26°