Lesson 5 4 Centroids Centroid Median Centroid Practice

Lesson 5. 4: Centroids

Centroid • Median: • Centroid:

Practice • D is the midpoint of AB. E is the midpoint of AC. F is the midpoint of BC. A Construct medians AF, BE, and CD. B C

Practice • What is the centroid of △ABC? • AF = 15. AG = ___ A GF = ___ • GC = 12 GD = ___ CD = ___ GE = 4 BG = ___ BE = ___ D E G B F C

Practice • AF = 24. AG = ___ GF = ___ • GC = 14 GD = ___ CD = ___ A GE = 9 BG = ___ BE = ___ D E G B F C

Application • Center of Balance:

Lesson 5. 5: Indirect Proof

Practice • Given: At least one answer choice is true. If x and y are positive numbers less than 100, then x + y = ___ a) b) c) d) e) -3 0 15 201 ∞

Indirect Proof • Elimination Proof: Given at least one true statement, show these statements are true by proving all others false. • Indirect Proof: To prove a statement is false, assume the statement to be true and show its truth leads to an impossible conclusion.

Proof Toolkit Triangle Angle-Sum Theorem: Substitution: Equilateral Triangle: Computation Properties:

Practice • Given: x + y = 5 and x ≠ 3. Prove: y ≠ 2.

Practice • Given: △ABC • Prove: An equilateral triangle cannot have a right angle. Statements 1) ______ 2) ∠A = 90 o 3) ∠A = ∠B = ∠C 4) ∠A + ∠B + ∠C = 180 o 5) ______ 6) 270 o ≠ 180 o Reasons Given Assume true. ____________ Substitution ______

Practice • Given: △ABC • Prove: A triangle cannot Statements have two right angles. 1) ______ 2) ∠A = 90 o, ∠B = 90 o 3) ∠A + ∠B + ∠C = 180 o 4) ______ 5) 180 o + ∠C = 180 o 6) ∠C = 0 o 7) ∠A ≠ 90 o, ∠B ≠ 90 o Reasons Given Assume true. ______ Substitution ____________ Angles ≠ 0 o

Extra Practice (if warranted) • Given: Two angles of ∆ABC are 50 o and 60 o. • Prove: ∆ABC is not a right triangle. - Assume ∆ABC is a right triangle. Then, 50 + 60 + ___ = 200 o. But, by the _____ a triangle can only have 180 o. This is a contradiction. Therefore, ∆ABC is _____.

Homework • Lesson 5. 4, #8 – 13, 17 – 18 Lesson 5. 5, #7 – 17 odd Quiz on Chapter 5 in two classes. • p. 345, #5 – 15