Bellringer Find the slope of the line passing through the given points. 1. A(9,6),B(8,12) -6 2. C(3,-2), D(0,6) 8/3 3. (-3,7), F(-3,12) undefined In chap.1 you learned the definition of midpoint of a segment. What do you think a midsegment of a triangle? A segment that connects the midpoints of 2 sides of the triangle.
5-1 The Mid-segment theorem and 5.7 The Hinge Theorem Geometry: Chapter 5 Relationship within Triangles 5-1 The Mid-segment theorem and 5.7 The Hinge Theorem
Review Chapter 3- We examined the angles formed by parallel lines and transversal lines. Chapter 4- We observed congruence in triangles. Chapter 1- We covered the essential tools of geometry Chapter 2-We explored the reasoning and logic of math.
Lesson’s Purpose Objective Essential Question What properties of midsegments can help you solve problems? Midsegments are parallel to the third side of a triangle, so you can solve problems using properties of parallel lines. Second, midsegments are half the length of the third segment, so you find missing side or segment lengths To use properties of midsegments to solve problems. To apply inequalities in two triangles
Mid-Segment Definition The mid-segment of a triangle (also called a midline) is a segment joining the midpoints of two sides of a triangle.
Mid-Segment Theorem The midsegment is parallel to the third side of the triangle. The length of the midsegment is half the length of the third side.
Practice Example QR is a midsegment of ∆MNO. What is the length of MO ? Step 1: Start by writing an equation using the Triangle Midsegment Theorem. ½ MO=QR Step 2: Substitute information: ½ MO=20 MO=2(20) Step 3: Solve-So, MO = 40.
Practice Examples AB is a midsegment of GEF. What is the value of x? Step 1: Set-up problem- 2AB = GF Step2: Substitute information: 2(2x) = 20 Step 3-Solve the equation:4x = 20 x = 5
Examples JK 9 GH 30 TU 13 Find the length of the in dedicated segments AC 30 JK 9 GH 30 TU 13
Examples In each triangle, AB is a midsegment. Find the value of x. 2(2x)=3x+11 4x=3x+11 X=11 2(3x)=5x+7 6x=5x+7 X=7 2(2x+5)=3x+15 4x+10=3x+15 X=5 X-7=2x-17 X=10 X=11 X=7 X=10
Hinge Theorem or SAS Inequality Theorem Consider ∆ABC and ∆XYZ. If , and mY . mB, then XZ > AC. This is the Hinge Theorem (SAS Inequality Theorem).
Practice Examples Which length is greater, GI or MN? Identify congruent sides: MO GH NO HI Compare included angles: mH > mO. By the Hinge Theorem, the side opposite the larger included angle is longer. So, GI > MN.
Examples 1.What is the inequality relationship between LP and XA in the figure at the right? XA>LP 2. At which time is the distance between the tip of a clock’s hour hand and the tip of its minute hand greater, 5:00 or 5:15? 5:00
Converse Hinge Theorem (SSS Inequality Theorem) Consider ∆LMN and ∆PQR. If , and PR > LN, then mQ > mM. This is the Converse of the Hinge Theorem (SSS Inequality Theorem).
Practice Examples Write an inequality: 72 > 5x + 2 by Converse of the Hinge Theorem 70 > 5x Subtract 2 from each side. 14 > x Divide each side by 5. So x<14. Write another inequality: mY > 0 The measure of an angle of a triangle is greater than 0. 5x + 2 > 0 Substitute. 5x > −2 So, x>-2/5 -2/5 <x>14 TR > ZX. What is the range of possible values for x? The triangles have two pairs of congruent sides, because RS = XY and TS = ZY . So, by the Converse of the Hinge Theorem, mS > mY.
Examples Find the range of possible values for each variable. How can you compare two triangles that have two pairs of congruent sides? You can compare two triangles that have two pairs of congruent sides using the Hinge Theorem.
Real World Connection
Summary Midsegments are parallel to third side of triangle Midsegments are half the length of third segment. Can solve with properties of parallel lines You can compare two triangles that have two pairs of congruent sides using the Hinge Theorem
Homework Sec. 5-1 pg.305-6 #'s 7-22 Sec. 5-7 Pg. 357 #'s 6-12 Ticket Out: How is the length of the midsegment of a triangle related to length of the third side? The length of the midsegment is half the length of the third side.