Pearson Unit 1 Topic 5: Relationships Within Triangles 5-2: Midsegments of Triangles Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle.

Investigation: finding the length of the midsegment

Investigation: the Midsegment Triangle

The midsegment triangle forms 4 congruent triangles.

Example: 1 BD = 8.5 Find each measure. BD ∆ Midsegment Thm. Substitute 17 for AE. BD = 8.5 Simplify.

Example: 2 Find each measure. m∡CBD ∆ Midsegment Thm. mCBD = mBDF Alt. Int. s Thm. mCBD = 26° Substitute 26° for mBDF.

Example: 3 2(36) = JL 72 = JL Find each measure. JL ∆ Midsegment Thm. Substitute 36 for PN and multiply both sides by 2. 72 = JL Simplify.

Example: 4 mMLK = mJMP mMLK = 102° Find each measure. m∡MLK ∆ Midsegment Thm. mMLK = mJMP Similar triangles mMLK = 102° Substitute.

Example: 5 The length of the support ST is 23 inches. In an A-frame support, the distance PQ is 46 inches. What is the length of the support ST if S and T are at the midpoints of the sides? ∆ Midsegment Thm. Substitute 46 for PQ. Simplify. ST = 23 The length of the support ST is 23 inches.

Example: 6 Anna measured the distance AE across the base of Capulin Volcano, an extinct volcano in New Mexico to be 1550 m. She measures a triangle at one side of the volcano as shown in the diagram. What is HF? ∆ Midsegment Thm. Substitute 1550 for AE. Simplify. HF = 775 m

Example: 7 Solve for x to find the length of the midsegment and the side of the triangle. 2(x + 3) = 5x – 9 2x + 6 = 5x – 9 6 = 3x – 9 15 = 3x 5 = x Midsegment = 5 + 3 = 8 Side of triangle = 5(5) – 9 = 25 – 9 = 16 x + 3 5x - 9

Example: 8 The vertices of ∆XYZ are X(–1, 8), Y(9, 2), and Z(3, –4). M and N are the midpoints of XZ and YZ. Show that and Step 1 Find the coordinates of M and N.

Example: 8 continued Step 2 Compare the slopes of MN and XY. Since the slopes are the same,

Example: 8 continued = 25+9 = 34 = 100+36 = 136 = 4 34 =2 34 Step 3 Compare the lengths of MN and XY. = 25+9 = 34 = 100+36 = 136 = 4 34 =2 34

Example: 9 The vertices of ΔRST are R(–7, 0), S(–3, 6), and T(9, 2). M is the midpoint of RT, and N is the midpoint of ST. Show that and Step 1 Find the coordinates of M and N.

Example: 9 continued Step 2 Compare the slopes of MN and RS. Since the slopes are equal .

Example: 9 continued = 9+4 = 13 = 36+16 = 52 = 4 13 =2 13 Step 3 Compare the lengths of MN and RS. = 9+4 = 13 = 36+16 = 52 = 4 13 =2 13 The length of MN is half the length of RS.

Example: 10 The midpoints of 3 sides of a triangle are K(-6, 2), L(-2, 4), and M(-4, 0). Find the coordinates of the vertices of the triangle. Hint: remember that the midsegment is parallel to the third side and half the length of the third side. Use slope to help you find the vertices of the triangle. Steps: Plot the given midpoints of the triangle. Find slope of each segment of triangle. Go to opposite vertex and do the slope: once forward, once backwards. Vertices: (-8, -2), (-4, 6), (0, 2).