Transcript Lesson
Lesson 5-2 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 medians in triangles • Identify and use 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 • 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 • Centroid – the point of concurrency for the medians of a triangle; point of balance for any triangle • Orthocenter – intersection point of the altitudes of a triangle; no special significance Theorems • 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. 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 Centroid Theorem Example 1 ALGEBRA Points U, V, and W are the midpoints of respectively. Find a, b, and c. Example 1a 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. Example 1b 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. Example 1c 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: Example 2 ALGEBRA Points T, H, and G are the midpoints of respectively. Find w, x, and y. Answer: 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 Orthocenter Special Segments in Triangles Name Type Point of Concurrency Center Special Quality Median segment Centroid Center of Gravity Altitude segment Orthocenter none From / To Vertex midpoint of segment Vertex none Location of Point of Concurrency Name Point of Concurrency Median Centroid Altitude Orthocenter Triangle Classification Acute Right Obtuse Inside Inside Inside Inside Vertex - 90 Outside Summary of Special Segments 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 Example 3 Identify each special segment in the triangle D Perpendicular bisector RM right angle at a midpoint Angle bisector DT FM from vertex to midpoint Altitude from vertex with right angle M S cuts angle in half Median R T ES E Summary & Homework • Summary: – Medians and altitudes of a triangle are all special segments in triangles – Altitudes form right angles – Medians go to midpoints • Centroid is the balance point • Located 2/3 the distance from the vertex • Homework: – pg 337-41; 1, 2, 5-10, 20, 26-30