Lesson 5 1 Bisectors Medians and Altitudes Transparency

Lesson 5 -1 Bisectors, Medians and Altitudes

Transparency 5 -1 5 -Minute Check on Chapter 4 Refer to the figure. 1. Classify the triangle as scalene, isosceles, or equilateral. 2. Find x if m A = 10 x + 15, m B = 8 x – 18, and m C = 12 x + 3. 3. Name the corresponding congruent angles if RST UVW. 4. Name the corresponding congruent sides if LMN OPQ. 5. Find y if DEF is an equilateral triangle and m F = 8 y + 4. 6. Standardized Test Practice: What is the slope of a line that contains (– 2, 5) and (1, 3)? A – 2/3 B 2/3 C – 3/2 D 3/2

Transparency 5 -1 5 -Minute Check on Chapter 4 Refer to the figure. 1. Classify the triangle as scalene, isosceles, or equilateral. isosceles 2. Find x if m A = 10 x + 15, m B = 8 x – 18, and m C = 12 x + 3. 6 3. Name the corresponding congruent angles if RST UVW. R U; S V; T W 4. Name the corresponding congruent sides if LMN OPQ. LM OP; MN PQ; LN OQ 5. Find y if DEF is an equilateral triangle and m F = 8 y + 4. 6. Standardized Test Practice: What is the slope of a line that contains (– 2, 5) and (1, 3)? A – 2/3 B 2/3 7 C – 3/2 D 3/2

Objectives • Identify and use perpendicular bisectors and angle bisectors in triangles • Identify and use medians and altitudes in triangles

Vocabulary • Concurrent lines – three or more lines that intersect at a common point • Point of concurrency – the intersection point of three or more lines • Perpendicular bisector – passes through the midpoint of the segment (triangle side) and is perpendicular to the segment • Median – segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex • Altitude – a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side

Vocabulary • Circumcenter – the point of concurrency of the perpendicular bisectors of a triangle; the center of the largest circle that contains the triangle’s vertices • Centroid – the point of concurrency for the medians of a triangle; point of balance for any triangle • Incenter – the point of concurrency for the angle bisectors of a triangle; center of the largest circle that can be drawn inside the triangle • Orthocenter – intersection point of the altitudes of a triangle; no special significance

Theorems • Theorem 5. 1 – Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. • Theorem 5. 2 – Any point equidistant from the endpoints of the segments lies on the perpendicular bisector of a segment. • Theorem 5. 3, Circumcenter Theorem – The circumcenter of a triangle is equidistant from the vertices of the triangle. • Theorem 5. 4 – Any point on the angle bisector is equidistant from the sides of the triangle. • Theorem 5. 5 – Any point equidistant from the sides of an angle lies on the angle bisector. • Theorem 5. 6, Incenter Theorem – The incenter of a triangle is equidistant from each side of the triangle. • Theorem 5. 7, Centroid Theorem – The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

Triangles – Perpendicular Bisectors A Note: from Circumcenter Theorem: AP = BP = CP Z Midpoint of AC Circumcenter Midpoint of AB X P C Y Midpoint of BC B Circumcenter is equidistant from the vertices

Triangles – Angle Bisectors A Note: from Incenter Theorem: QX = QY = QZ Z Q Incenter C X Y B Incenter is equidistant from the sides

A Triangles – Medians Note: from Centroid theorem BM = 2/3 BZ Midpoint Z of AC Midpoint of AB X M Centroid Median from B C Y Midpoint of BC B Centroid is the point of balance in any triangle

A Triangles – Altitudes Note: Altitude is the shortest distance from a vertex to the line opposite it Z Altitude from B C ter n e c tho Or X Y B Orthocenter has no special significance for us

Special Segments in Triangles Name Type Point of Concurrency Center Special Quality From / To Equidistant from vertices None midpoint of segment Incenter Equidistant from sides Vertex none Vertex midpoint of segment Perpendicular Line, Circumcenter bisector segment or ray Angle bisector Line, segment or ray Median segment Centroid Center of Gravity Altitude segment Orthocenter none Vertex none

Location of Point of Concurrency Name Point of Concurrency Perpendicular bisector Circumcenter Triangle Classification Acute Right Obtuse Inside hypotenuse Outside Angle bisector Incenter Inside Median Centroid Inside Altitude Orthocenter Inside Vertex - 90 Outside

Given: Find: m DGE Proof: Statements Reasons 1. Given 2. 3. 4. 5. 2. Angle Sum Theorem 3. Substitution 4. Subtraction Property 5. Definition of angle bisector 6. Angle Sum Theorem 7. Substitution 8. Subtraction Property 6. 7. 8.

Given: . Find: m ADC Proof: Statements. Reasons 1. Given 2. 3. 4. 5. 2. Angle Sum Theorem 3. Substitution 4. Subtraction Property 5. Definition of angle bisector 6. Angle Sum Theorem 7. Substitution 8. Subtraction Property 6. 7. 8.

ALGEBRA Points U, V, and W are the midpoints of respectively. Find a, b, and c. Find a. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 14. 8 from each side. Divide each side by 4.

Find b. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 6 b from each side. Subtract 6 from each side. Divide each side by 3.

Find c. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 30. 4 from each side. Divide each side by 10. Answer:

ALGEBRA Points T, H, and G are the midpoints of respectively. Find w, x, and y. Answer:

Summary & Homework • Summary: – Perpendicular bisectors, angle bisectors, medians and altitudes of a triangle are all special segments in triangles – Perpendiculars and altitudes form right angles – Perpendiculars and medians go to midpoints – Angle bisector cuts angle in half • Homework: – Day 1: pg 245: 46 -49 – Day 2: pg 245: 51 -54