The Triangle Midsegment Theorem Geometry The Triangle Midsegment Theorem CONFIDENTIAL

Warm up Find each length. 1) NX 2) MR 3) NP P 1) S 3 R X 4.5 5.5 M Q N CONFIDENTIAL

The Triangle Midsegment Theorem A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegment, which form the midsegment triangle. Q Y X P R Z CONFIDENTIAL

Examining Midsegments in the coordinate Plane y H(-5, 6) 6 L 4 J(1, 2) K 2 x -2 -2 G(-7, -2) Next Page: CONFIDENTIAL

y H(-5, 6) 6 L 4 J(1, 2) K 2 x -2 -2 G(-7, -2) Next Page: CONFIDENTIAL

y H(-5, 6) 6 L 4 J(1, 2) K 2 x -2 -2 G(-7, -2) CONFIDENTIAL

Now you try! CONFIDENTIAL

The relationship show in Example 1 is true for the three midsegments of every triangle . CONFIDENTIAL

Triangle Midsegment Theorem A Midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. B A D E C CONFIDENTIAL

Using the triangle Midsegment Theorem Find each measure. S U A) UW 5.2 R 41˚ ∆ Midsegment Thm. Substitute 7.4 for ST. Simplify. V 7.4 W T Next Page: CONFIDENTIAL

S B) m/ SVU U 5.2 R 41˚ V 7.4 ∆ Midsegment Thm. Alt. Int. /s Thm. Substitute 41˚ for m/ VUW. W T CONFIDENTIAL

Now you try! 2) Find each measure. a) JL b) PM c) m/ MLK P K J 36 102˚ 97 L CONFIDENTIAL

Indirect Measurement Application Anna wants to find the distance across the base of Capulin Volcano, an extinct volcano in New Mexico. She measures a triangle at one side of the volcano as shown in the diagram. What is AE? C ∆ midsegment Thm. Substitute 775 foe BD. Multiply both sides by 2. 920 m 700 m 775 m B D 920 m 700 m E A The distance AE across the base of the volcano is about 1550 meters. CONFIDENTIAL

Now you try! 3) Suppose Anna’s result in Example 3 is correct to check it, she measures a second triangle. How many meters will she measure between H and F? H F A E G 640 m 1005 m CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

Assessment CONFIDENTIAL

Find each measure. 2) NM 3) XZ 4) NZ 5) m/ LMN 6) m/ YXZ 7) m/ XLM Y 10.2 5.6 29˚ CONFIDENTIAL

8) In this A-frame house, the width of the first floor XZ is 30 feet. The second floor CD is slightly above and parallel to the midsegment of ∆XYZ. Is the width of the second floor more or less than 5 yards? Explain. Y C D X Z CONFIDENTIAL

The Triangle Midsegment Theorem Let’s review The Triangle Midsegment Theorem A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegment, which form the midsegment triangle. Q Y X P R Z CONFIDENTIAL

Examining Midsegments in the coordinate Plane y H(-5, 6) 6 L 4 J(1, 2) K 2 x -2 -2 G(-7, -2) Next Page: CONFIDENTIAL

y H(-5, 6) 6 L 4 J(1, 2) K 2 x -2 -2 G(-7, -2) Next Page: CONFIDENTIAL

y H(-5, 6) 6 L 4 J(1, 2) K 2 x -2 -2 G(-7, -2) CONFIDENTIAL

The relationship show in Example 1 is true for the three midsegments of every triangle . CONFIDENTIAL

Triangle Midsegment Theorem A Midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. B D E A C CONFIDENTIAL

Using the triangle Midsegment Theorem Find each measure. S U A) UW 5.2 R 41˚ ∆ Midsegment Thm. Substitute 7.4 for ST. Simplify. V 7.4 W T Next Page: CONFIDENTIAL

S B) m/ SVU U 5.2 R 41˚ V 7.4 ∆ Midsegment Thm. Alt. Int. /s Thm. Substitute 41˚ for m/ VUW. W T CONFIDENTIAL

Indirect Measurement Application Anna wants to find the distance across the base of Capulin Volcano, an extinct volcano in New Mexico. She measures a triangle at one side of the volcano as shown in the diagram. What is AE? C ∆ midsegment Thm. Substitute 775 foe BD. Multiply both sides by 2. 920 m 700 m 775 m B D 920 m 700 m E The distance AE across the base of the volcano is about 1550 meters. A CONFIDENTIAL

You did a great job today! CONFIDENTIAL