Geometry The Triangle Midsegment Theorem

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1)

Warm up

Find each length.

1) NX 2) MR 3) NP P M S 3 X R

4.5

Q N CONFIDENTIAL 2

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 midsegement: XY, YZ, ZX X Y midsegement triangle: XYZ P Z CONFIDENTIAL R 3

Examining Midsegments in the coordinate Plane

In GHJ, show that midsegment KL is parallel to GJ and that KL = 1 2 GJ.

Step 1 Find the coordinate of K and L.

 midpt. of GH = -7 + (-5) 2 , -2 + 6 2  = (-6, 2 )  mdpt. of HJ = -5 + 1 2 , 6+2 2  = (-2,4) CONFIDENTIAL y H(-5, 6) K L 6 4 J(1, 2) 2 -2 G(-7, -2) 0 -2

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x 4

Step 2 Compare the slopes of KL and GJ.

4 - 2 slope of KL = -2-(-6) 1 = 2 2-(-2) slope of GJ = 1-(-7) 1 = 2 Since the slopes are the same, KL || GJ.

CONFIDENTIAL y H(-5, 6) K L 6 4 J(1, 2) 2 -2 G(-7, -2) 0 -2 x

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Step 3 Compare the lengths of KL and GJ.

KL = [-2-(-6)] 2 + (4-2) 2 = 2 5 GJ = [1-(-7)] 2 + [(2-(-2)] 2 =4 5 Since 2 5 = 1 2 (4 5), KL = 1 2 GJ.

CONFIDENTIAL y H(-5, 6) K L 6 4 J(1, 2) 2 -2 G(-7, -2) 0 -2 x 6

Now you try!

1) The vertices of RST are R(-7, 0), S(-3, 6), and T(9, 2).M is the midpoint of RT, and N is the 1 midpoint of ST.Show that MN RS and MN = 2 RS.

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

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Triangle Midsegment Theorem Theorem 4.1

A Midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.

DE AC, DE = 1 2 AC A D B E CONFIDENTIAL C 9

Using the triangle Midsegment Theorem

Find each measure.

A) UW UW = 1 2 ST UW = 1 2 (7.4) UW = 3.7

∆ Midsegment Thm.

Substitute 7.4 for ST.

Simplify.

R U 41˚ 5.2

S V 7.4

W CONFIDENTIAL T

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B) m/ SVU UW ST m  SVU = m  VUW m  SVU = 41  R ∆ Midsegment Thm.

Alt. Int. /s Thm.

Substitute 41˚ for m/ VUW.

U 41˚ 5.2

S V 7.4

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Now you try!

2) Find each measure.

a) 72 b) 43.5

c)102 0 a) JL b) PM c) m/ MLK J 102˚ M L P 36 N CONFIDENTIAL K 12

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?

BD = 1 2 AE 775 = 1 2 AE 1550 = AE C ∆ midsegment Thm.

Substitute 775 foe BD.

700 m B 775 m 920 m D Multiply both sides by 2.

700 m 920 m A The distance AE across the base of the volcano is about 1550 meters.

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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?

3) A 640 m H 640 m E G F 1005 m 1005 m CONFIDENTIAL 14

Now some problems for you to practice !

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Assessment

1) The vertices of PQR are P(-4, -1), Q(2,9), and R(6,3). S is the midpoint of PQ, and T is the midpoint of QR. Show that ST PR 1 and ST = 2 PR.

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Find each measure.

2) 5.1

3) 11.2

4) 5.6

5)29 0 6)29 0 7)29 0 X 2) NM 3) XZ 4) NZ 5) m/ LMN 6) m/ YXZ 7) m/ XLM Y 10.2

L N CONFIDENTIAL 5.6

29˚ M Z 17

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 X CONFIDENTIAL D Z 18

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 midsegement: XY, YZ, ZX X Y midsegement triangle: XYZ P Z CONFIDENTIAL R 19

Examining Midsegments in the coordinate Plane

In GHJ, show that midsegment KL is parallel to GJ and that KL = 1 2 GJ.

Step 1 Find the coordinate of K and L.

 midpt. of GH = -7 + (-5) 2 , -2 + 6 2  = (-6, 2 )  mdpt. of HJ = -5 + 1 2 , 6+2 2  = (-2,4) CONFIDENTIAL y H(-5, 6) K L 6 4 J(1, 2) 2 -2 G(-7, -2) 0 -2

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x 20

Step 2 Compare the slopes of KL and GJ.

4 - 2 slope of KL = -2-(-6) 1 = 2 2-(-2) slope of GJ = 1-(-7) 1 = 2 Since the slopes are the same, KL || GJ.

CONFIDENTIAL y H(-5, 6) K L 6 4 J(1, 2) 2 -2 G(-7, -2) 0 -2 x

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Step 3 Compare the lengths of KL and GJ.

KL = [-2-(-6)] 2 + (4-2) 2 = 2 5 GJ = [1-(-7)] 2 + [(2-(-2)] 2 =4 5 Since 2 5 = 1 2 (4 5), KL = 1 2 GJ.

CONFIDENTIAL y H(-5, 6) K L 6 4 J(1, 2) 2 -2 G(-7, -2) 0 -2 x 22

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

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Theorem 4.1

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 DE AC, DE = 1 2 AC D E A C CONFIDENTIAL 24

Using the triangle Midsegment Theorem

Find each measure.

A) UW UW = 1 2 ST UW = 1 2 (7.4) UW = 3.7

∆ Midsegment Thm.

Substitute 7.4 for ST.

Simplify.

R U 41˚ 5.2

S V 7.4

W T CONFIDENTIAL

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B) m/ SVU UW ST m  SVU = m  VUW m  SVU = 41  R ∆ Midsegment Thm.

Alt. Int. /s Thm.

Substitute 41˚ for m/ VUW.

U 41˚ 5.2

S V 7.4

W T CONFIDENTIAL 26

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 BD = 1 2 AE 775 = 1 2 AE 1550 = AE ∆ midsegment Thm.

Substitute 775 foe BD.

Multiply both sides by 2.

700 m B 775 m 700 m 920 m D 920 m E The distance AE across the base of the volcano is about 1550 meters.

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You did a great job today!

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