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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 mA = 10x + 15, mB = 8x – 18, and mC = 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 mF = 8y + 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 mA = 10x + 15, mB = 8x – 18, and mC = 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 mF = 8y + 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 Centroid M C Median from B 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 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 Inside Inside Median Centroid Inside Inside Inside Altitude Orthocenter Inside Vertex - 90 Outside Given: Find: mDGE Proof: Statements Reasons 1. 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: mADC Proof: Statements. Reasons 1. 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 6b 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