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Lesson 5-1

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.

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Lesson 5-1

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  1. Lesson 5-1 Bisectors, Medians and Altitudes

  2. 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 = 10x + 15, mB = 8x – 18, and mC = 12x + 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 = 8y + 4. 6. What is the slope of a line that contains (–2, 5) and (1, 3)? Standardized Test Practice: 2/3 –3/2 A B C D 3/2 –2/3

  3. 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 = 10x + 15, mB = 8x – 18, and mC = 12x + 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 = 8y + 4. 7 6. What is the slope of a line that contains (–2, 5) and (1, 3)? Standardized Test Practice: 2/3 –3/2 A B C D 3/2 –2/3

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

  5. 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

  6. 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

  7. 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.

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

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

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

  11. Triangles – Altitudes A Note: Altitude is the shortest distance from a vertex to the line opposite it Z Altitudefrom B C Orthocenter X Y B Orthocenter has no special significance for us

  12. Special Segments in Triangles

  13. Location of Point of Concurrency

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

  15. Given: Proof: . Reasons Statements. 1. Given2. Angle Sum Theorem3. Substitution4. Subtraction Property 5. Definition of anglebisector6. Angle Sum Theorem7. Substitution8. Subtraction Property 1. Given 1.2.3.4.5.6.7.8. mADC Find:

  16. 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.

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

  18. 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:

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

  20. 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

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