5.1

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Midsegment Theorem and Coordinate Proof

5.1 Warm-Up

In Exercises 1– 4, use A(0, 10), B(24, 0), and C(0, 0).

1. Find AB.

ANSWER 26

ANSWER (0, 5)

2. Find the midpoint of CA.

5.1 Warm-Up

In Exercises 1– 4, use A(0, 10), B(24, 0), and C(0, 0).

ANSWER (12, 5)

4. Find the slope of AB.

3. Find the midpoint of AB.

ANSWER – 512

5.1 Example 1

SOLUTION

UV =12 RT =

12 ( 90 in.) = 45 in.

RS = 2 VW = 2 ( 57 in.) = 114 in.

Triangles are used for strength in roof trusses. In the diagram, UV and VW are midsegments of RST. Find UV and RS.

CONSTRUCTION

5.1 Guided Practice

1. Copy the diagram in Example 1. Draw and name the third midsegment.

2. In Example 1, suppose the distance UW is 81 inches. Find VS.

ANSWER 81 in.

ANSWER UW

5.1 Example 2

In the kaleidoscope image, AE BE and AD CD . Show that CB DE .

SOLUTION

Because AE BE and AD CD , E is the midpoint of AB and D is the midpoint of AC by definition.

Then DE is a midsegment of ABC by definition and CB DE by the Midsegment Theorem.

5.1 Example 3

Place each figure in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex.

a. Rectangle: length is h and width is k

b. Scalene triangle: one side length is d

SOLUTION

It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis.

5.1 Example 3

a. Let h represent the length and k represent the width.

b. Notice that you need to use three different variables.

5.1 Guided Practice

3. In Example 2, if F is the midpoint of CB , what do you know about DF ?

ANSWER

DF AB and DF is half the length of AB.

DF is a midsegment of ABC.

5.1 Guided Practice

4. Show another way to place the rectangle in part (a) of Example 3 that is convenient for finding side lengths. Assign new coordinates.

ANSWER

5.1 Guided Practice

5. Is it possible to find any of the side lengths in part (b) of Example 3 without using the Distance Formula? Explain.

Yes; the length of one side is d.ANSWER

6. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.

ANSWER (m, m)

5.1 Example 4

SOLUTION

Place an isosceles right triangle in a coordinate plane. Then find the length of the hypotenuse and the coordinates of its midpoint M.

Place PQO with the right angle at the origin. Let the length of the legs be k. Then the vertices are located at P(0, k), Q(k, 0), and O(0, 0).

5.1 Example 4

Use the Distance Formula to find PQ.

PQ = (k – 0) + (0 – k)22 = k + (– k)

2 2 = k + k2 2

= 2k 2 = k 2

Use the Midpoint Formula to find the midpoint M of the hypotenuse.

M( )0 + k , k + 02 2 = M( , )k

2k2

5.1 Example 5

GIVEN: DE is a midsegment of OBC.

PROVE: DE OC and DE = OC12

Write a coordinate proof of the Midsegment Theorem for one midsegment.

SOLUTION

STEP 1 Place OBC and assign coordinates. Because you are finding midpoints, use 2p, 2q, and 2r. Then find the coordinates of D and E.

D( )2q + 0, 2r + 02 2

= D(q, r) E( )2q + 2p, 2r + 02 2

= E(q+p, r)

5.1 Example 5

STEP 2 Prove DE OC . The y-coordinates of D and E are the same, so DE has a slope of 0. OC is on the x-axis, so its slope is 0.

STEP 3 Prove DE = OC. Use the Ruler Postulate12

to find DE and OC .

DE =(q + p) – q = p OC = 2p – 0 = 2p

Because their slopes are the same, DE OC .

So, the length of DE is half the length of OC

5.1 Guided Practice

7. In Example 5, find the coordinates of F, the midpoint of OC . Then show that EF OB .

(p, 0); slope of EF = = ,

slope of OB = = , the slopes of

EF and OB are both , making EF || OB.

r 0(q + p) p q

r

2r 02q 0 q

r

qr

ANSWER

5.1 Guided Practice

8. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJ a right triangle? Find the side lengths and the coordinates of the midpoint of each side.

ANSWER

yes; OJ = m, JH = n,

HO = m2 + n2,

OJ: ( , 0), JH: (m, ),

HO: ( , )

2m

2n

2m

2n

Sample:

5.1 Lesson Quiz

Use the figure for Exercises 1–4.

1. If UV = 13, find RT.

2. If ST = 20, find UW.

ANSWER 26

ANSWER 10

5.1

3. If the perimeter of RST = 68 inches, find the perimeter of UVW.

Lesson Quiz

Use the figure for Exercises 1–4.

ANSWER 34 in.

4. If VW = 2x – 4, and RS = 3x – 3, what is VW?

ANSWER 6

5.1 Lesson Quiz

5. Place a rectangle in a coordinate plane so itsvertical side has length a and its horizontal sidehas width 2a. Label the coordinates of eachvertex.

ANSWER