UNIT 6 Properties and Attributes of Triangles 5

UNIT 6 Properties and Attributes of Triangles

5. 1 Perpendicular and Angle Bisectors Essential Question Compare and Contrast Perpendicular Bisectors and Angle bisectors.

Vocabulary Perpendicular Bisector Circumcenter Angle Bisector Incenter

Perpendicular Bisector

Example 1 A: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. MN MN = LN Bisector Thm. MN = 2. 6 Substitute 2. 6 for LN.

Example 1 B: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. BC Since AB = AC and , is the perpendicular bisector of by the Converse of the Perpendicular Bisector Theorem. BC = 2 CD BC = 2(12) = 24 Definition of Perpendicular bisector. Substitute 12 for CD.

Example 1 C: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. TU TU = UV 3 x + 9 = 7 x – 17 9 = 4 x – 17 26 = 4 x 6. 5 = x Bisector Thm. Substitute the given values. Subtract 3 x from both sides. Add 17 to both sides. Divide both sides by 4. So TU = 3(6. 5) + 9 = 28. 5.

HINT Helpful Hint The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex.

Perpendicular Bisectors intersect at the Circumcenter When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.

The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

Example 1: Using Properties of Perpendicular Bisectors DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC. G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of ∆ABC. GC = CB GC = 13. 4 Circumcenter Thm. Substitute 13. 4 for GB.

Check It Out! Example 1 a Use the diagram. Find GM. MZ is a perpendicular bisector of ∆GHJ. GM = MJ GM = 14. 5 Circumcenter Thm. Substitute 14. 5 for MJ.

Check It Out! Example 1 b Use the diagram. Find GK. KZ is a perpendicular bisector of ∆GHJ. GK = KH GK = 18. 6 Circumcenter Thm. Substitute 18. 6 for KH.

Check It Out! Example 1 c Use the diagram. Find JZ. Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ. JZ = GZ Circumcenter Thm. JZ = 19. 9 Substitute 19. 9 for GZ.

Angle Bisectors

Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle.

Example 2 A: Applying the Angle Bisector Theorem Find the measure. BC BC = DC Bisector Thm. BC = 7. 2 Substitute 7. 2 for DC.

Example 2 B: Applying the Angle Bisector Theorem Find the measure. m EFH, given that m EFG = 50°. Since EH = GH, and , bisects EFG by the Converse of the Angle Bisector Theorem. Def. of bisector Substitute 50° for m EFG.

Example 2 C: Applying the Angle Bisector Theorem Find m MKL. Since, JM = LM, and , bisects JKL by the Converse of the Angle Bisector Theorem. m MKL = m JKM Def. of bisector 3 a + 20 = 2 a + 26 a + 20 = 26 a=6 Substitute the given values. Subtract 2 a from both sides. Subtract 20 from both sides. So m MKL = [2(6) + 26]° = 38°

A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle.

Remember! The distance between a point and a line is the length of the perpendicular segment from the point to the line.

Unlike the circumcenter, the incenter is always inside the triangle.

The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.

Example 3 A: Using Properties of Angle Bisectors MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN. P is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN. The distance from P to LM is 5. So the distance from P to MN is also 5.

Example 3 B: Using Properties of Angle Bisectors MP and LP are angle bisectors of ∆LMN. Find m PMN. m MLN = 2 m PLN PL is the bisector of MLN. m MLN = 2(50°) = 100° Substitute 50° for m PLN. m MLN + m LNM + m LMN = 180° Δ Sum Thm. 100 + 20 + m LMN = 180 Substitute the given values. m LMN = 60° Subtract 120° from both sides. PM is the bisector of LMN. Substitute 60° for m LMN.

5. 1 Perpendicular and Angle Bisectors Summary Compare and Contrast Perpendicular Bisectors and Angle bisectors. The perpendicular bisector______. Perpendicular bisectors intersect at ____ point of concurrency. The angle bisector______. Angle bisectors intersect at ____ point of concurrency. Both the perpendicular bisector and angle bisector ______________.