Coordinate Proofs

389

Student Journal

358

11.3

Coordinate Proofs

For use with Exploration 11.3

Name _________________________________________________________ Date _________

Essential Question How can you use a coordinate plane to write a

proof?

Go to BigIdeasMath.com for an interactive tool to investigate this exploration.

Work with a partner.

a. Use dynamic geometry software

to draw

AB with endpoints A(0, 0)

and B(6, 0).

b. Draw the vertical line

3.=x

c. Draw

A

BC

so that C lies on the

line

3.x =

d. Use your drawing to prove that

A

BC

is an

isosceles triangle.

Go to BigIdeasMath.com for an interactive tool to investigate this exploration.

Work with a partner.

a. Use dynamic geometry software to draw AB with endpoints A(0, 0) and

B(6, 0).

b. Draw the vertical line

3.x =

c. Plot the point C(3, 3) and draw

.

A

BC

Then use your drawing to prove that

A

BC

is an isosceles right triangle.

1 EXPLORATION: Writing a Coordinate Proof

2 EXPLORATION: Writing a Coordinate Proof

Sample

Points

A(0, 0)

B(6, 0)

C(3, y)

Segments

6AB =

Line

3x =

0

1

2

3

4

−1

01 2

A

C

B

3456

390

359

Name _________________________________________________________ Date __________

d. Change the coordinates of C so that C lies below the x-axis and

A

BC

is an

isosceles right triangle.

e. Write a coordinate proof to show that if C lies on the line

3=x

and

A

BC

is

an isosceles right triangle, then C must be the point (3, 3) or the point found in

part (d).

Communicate Your Answer

3. How can you use a coordinate plane to write a proof?

4. Write a coordinate proof to prove that

A

BC

with vertices A(0, 0), B(6, 0), and

()

33, 3C

is an equilateral triangle.

11.3

Coordinate Proofs (continued)

Sample

Points

A(0, 0)

B(6, 0)

C(3, 3)

Segments

6AB =

4.24BC =

4.24AC =

Line

3x =

2 EXPLORATION: Writing a Coordinate Proof (continued)

0

1

2

3

4

−1

01 2

A

C

B

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Student Journal

360

11.3

For use after Lesson 11.3

Name _________________________________________________________ Date _________

Notes:

Practice

Worked-Out Examples

Example #1

x

y

4

5

6

7

8

2

3

1

4

5 6 7 8 9321

O(0, 0)

D(9, 0)

C(0, 7)

CD =

√

——

(0 − 9)

2

+ (7 − 0)

2

=

√

—

(−9)

2

+ 7

2

=

√

—

81 + 49 =

√

—

130 ≈ 11.4

The length of the hypotenuse is about 11.4 units.

Write a plan for the proof.

Example #2

Place the ﬁgure in a coordinate plane and ﬁnd the indicated length.

x

y

4

2

8

M(8, 4)P(3, 4)

O(0, 0) N(5, 0)

Given Coordinates of vertices of

OPM

and



Prove

OPM

and



are isosceles triangles.

A right triangle with the leg lengths of 7 and 9 units; Find the length of the hypotenuse

ONM

—

≅

—

and

—

≅

—

OP PM MN O N

Find the lengths of , , to show that

—

OP

PM MN ON

, and

.

392

361

Name _________________________________________________________ Date __________

Extra Practice

In Exercises 1 and 2, place the figure in a coordinate plane in a convenient way.

Assign coordinates to each vertex. Explain the advantages of your placement.

1. an obtuse triangle with height of 3 units

and base of 2 units

2. a rectangle with length of 2w

In Exercises 3 and 4, write a plan for the proof.

3. Given Coordinates of vertices of

OPR

and

QRP

Proof

OPR QRP≅

11.3

x

y

O(0, 0)

P(2, 5)

Q(9, 5)

R(7, 0)

4

2

4

6

Practice A

Practice (continued)

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Student Journal

362

Name _________________________________________________________ Date _________

4. Given Coordinates of vertices of

OAB

and

CDB

Prove B is the midpoint of

A

D and

.OC

5. Graph the triangle with vertices A(0, 0), B(3m, m), and

C(0, 3m). Find the length and the slope of each side of the

triangle. Then find the coordinates of the midpoint of each

side. Is the triangle a right triangle? isosceles? Explain.

(Assume all variables are positive.)

6. Write a coordinate proof.

Given Coordinates of vertices of

OEF

and

OGF

Prove

OEF OGF≅

11.3

x

y

O(0, 0)

C(6, 6)

A(0, 4)

D(6, 2)

4 6 8

2

4

6

B

x

y

x

y

E(k, h)

G(2k, h)

O(0, 0)

F(0, 4h)

Practice (continued)

394

423

12.8

Practice B

Name _________________________________________________________ Date __________

In Exercises 1–3, place the figure in a coordinate plane in a convenient way.

Assign coordinates to each vertex. Explain the advantages of your placement.

1.

a rectangle twice as long as it is wide

2. a right triangle with a leg length of 3 units and a hypotenuse with a

positive slope

3. an obtuse scalene triangle

In Exercises 4 and 5, graph the triangle with the given vertices. Find the

length and the slope of each side of the triangle. Then find the coordinates

of the midpoint of each side. Is the triangle a right triangle? isosceles?

Explain.

4.

( ) ( )( )

0, 0 , , , 2 , 0J Kab L a

5.

( )( )( )

0, 0 , 5 , 0 , 8 , 4P Qa Ra a

In Exercises 6 and 7, find the coordinates of any unlabeled vertices. Then find

the indicated lengths.

6.

Find GH and FH. 7. Find BC and CD.

8. The vertices of a quadrilateral are given by the coordinates

( )( )

3, 5 , 5, 0 ,WX

( ) ()

3, 4 , and 5, 1 .YZ−− −

Is the quadrilateral a parallelogram? a trapezoid?

Explain your reasoning.

9. Write a coordinate proof for the following statement.

Any

A

BC formed so that vertex C is on the perpendicular bisector

of

AB is an isosceles triangle.

k units

J

F

G

H

x

y

2k units

A ED(2k, 0)

BC

x

y

Practice B