HSM12CC_GM_06_09_CM

6-9
Proofs Using
Coordinate Geometry
Vocabulary
Review
1. Circle the Midpoint Formula for a segment in the coordinate plane. Underline the
Distance Formula for a segment in the coordinate plane.
M5 ¢
x1 1 x2 y1 1 y2
,
≤
2
2
d 5 "(x2 2 x1)2 1 (y2 2 y1)2
y 2y
m 5 x2 2 x1
2
1
2. Circle the Midpoint Formula for a segment on a number line. Underline the
Distance Formula for a segment on a number line.
M5
x1 1 x2
2
d 5 |x1 2 x2|
m5
x1 2 x2
2
Vocabulary Builder
VEHR
x and y are
often used as
variables.
ee uh bul
Related Words: vary (verb), variable (adjective)
Definition: A variable is a symbol (usually a letter) that represents one
or more numbers.
Math Usage: A variable represents an unknown number in equations
and inequalities.
Use Your Vocabulary
Underline the correct word to complete each sentence.
3. An interest rate that can change is a variable / vary interest rate.
4. You can variable / vary your appearance by changing your hair color.
5. The amount of daylight variables / varies from summer to winter.
6. Circle the variable(s) in each expression below.
3n
41x
p2 2 2p
4
y
7. Cross out the expressions that do NOT contain a variable.
21m
Chapter 6
36 4 (2 ? 3)
9a2 2 4a
178
8 2 (15 4 3)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
variable (noun)
Problem 1 Writing a Coordinate Proof
Got It? Reasoning You want to prove that the midpoint of the hypotenuse of a
right triangle is equidistant from the three vertices. What is the advantage of using
coordinates O(0, 0), E(0, 2b), and F(2a, 0) rather than O(0, 0), E(0, b), and F(a, 0)?
8. Label each triangle.
y
E(0,
y
)
, 2b)
E(
M
M
x
O (0, 0)
F(a,
x
O (0, 0)
)
9. Use the Midpoint Formula M 5 ¢
M in each triangle.
x1 1 x2 y1 1 y2
,
≤ to find the coordinates of
2
2
Fisrt Triangle
a
a10
2 ,
, 0)
F(
Second Triangle
1
2
a
b 5 a 2,
2
b
a
1
,
2
0 1 2b
2 b 5(
, b)
10. Use the Distance Formula, d 5 "(x2 2 x1)2 1 (y2 2 y1)2 and your answers to
Exercise 9 to verify that EM 5 FM 5 OM for the first triangle.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
EM
FM
OM
11. Use the Distance Formula, d 5 "(x2 2 x1)2 1 (y2 2 y1)2 and your answers to
Exercise 9 to verify that EM 5 FM 5 OM for the second triangle.
EM
FM
OM
12. Which set of coordinates is easier to use? Explain.
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_______________________________________________________________________
179
Lesson 6-9
Writing a Coordinate Proof
Problem 2
Got It? Write a coordinate proof of the Triangle
y
Midsegment Theorem (Theorem 5-1).
B(
,
)
Given: E is the midpoint of AB and
F is the midpoint of BC
,
F(
Prove: EF 6 AC, EF 5 12AC
Use the diagram at the right.
)
+
E(
,
)
x
13. Label the coordinates of point C.
O C(
,
)
,
)
b 5(
,
A(
14. Reasoning Why should you make the coordinates
of A and B multiples of 2?
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_______________________________________________________________________
15. Label the coordinates of A and B in the diagram.
16. Use the Midpoint Formula to find the coordinates of E and F. Label the
coordinates in the diagram.
a
coordinates of F
+
+
,
2
b 5(
2
,
)
a
+
2
+
,
2
17. Use the Slope Formula to determine whether EF 6 AC.
2
slope of EF 5
5
2
2
slope of AC 5
5
2
18. Is EF 6 AC? Explain.
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_______________________________________________________________________
1
19. Use the Distance Formula to determine whether EF 5 2 AC.
EF 5
Å
AC 5
Å
(
20. 12 AC 5 12 ?
Chapter 6
2
(
2
)2 1 (
)2 1 (
5
2
2
)2 5 Î (a)2 1 (0)2 5
)2 5 Î (2a)2 1 (0)2 5
5 EF
180
)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
coordinates of E
Lesson Check • Do you know HOW?
Use coordinate geometry to prove that the diagonals of a rectangle are congruent.
y
21. Draw rectangle PQRS with P at (0, 0).
22. Label Q(a,
), R(
, b), and S(
,
).
23. Complete the Given and Prove statements.
Given: PQRS is a
x
Prove: PR >
.
24. Use the Distance Formula to find the length of eatch diagonal.
PR 5
QS 5
Ä
(
2
)2 1 (
2
)2 5
Ä
(
2
)2 1 (
2
)2 5
25. PR 5
, so PR >
.
Lesson Check • Do you UNDERSTAND?
y
Error Analysis Your classmate places a trapezoid on the coordinate
plane. What is the error?
P(b, c) Q(a − b, c)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
26. Check whether the coordinates are for an isosceles trapezoid.
OP 5
QR 5
Ä
Ä
(b 2
)2 1 (c 2
(a 2
x
O
)2 5
)2 1 (0 2
R(a, 0)
)2 5
27. Does the trapezoid look like an isosceles triangle?
Yes / No
28. Describe your classmate’s error.
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Math Success
Check off the vocabulary words that you understand.
proof
theorem
coordinate plane
coordinate geometry
Rate how well you can prove theorems using coordinate geometry.
Need to
review
0
2
4
6
8
Now I
get it!
10
181
Lesson 6-9