…BLM 2–1 Get Readypjk.scripts.mit.edu/lab/2d/Chapter_2_ALL.pdf · BLM 2–8 Section 2.4...

Name: ___________________________________ Date: _______________________________

…BLM 2–1...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 2–1 Get Ready

Get Ready

Solving Equations 1. Solve each equation.

a) 4x + 3 = 11 b) 8y − 5 = 6y + 7

c) 1 13 55 4z z+ = + d) 5d =

2. Write each equation in the form y = mx + b.

a) x − y + 3 = 0 b) 5x + y − 7 = 0

c) 3x + 6y − 8 = 0 d) 1 5 2 03 x y− + =

Slope of a Line 3. Find the slope of the line through each pair of

points. a) (− 4, −6) and (−6, 10) b) (5, 2) and (8, −3) c) (3, 8) and (5, 12) d) (7, −3) and (−1, 5)

Equation for a Line 4. Find an equation for the line that

a) has slope − 4 and y-intercept 7

b) has slope 14− and y-intercept −3

c) has slope 5 and passes through (4, 3)

d) has slope 13 and passes through (−2, 5)

5. Find an equation for the line that passes

through each pair of points. a) (1, 3) and (5, 11) b) (−2, 4) and (3, −1) c) (−5, 2) and (−3, − 4) d) (−3, −10) and (1, 2)

Parallel and Perpendicular Lines 6. Find the slope of a line with each property.

a) parallel to the line defined by y = 5x + 4

b) parallel to the line defined by y = 14 x − 2

c) perpendicular to the line defined by y = 3x + 5

d) perpendicular to the line defined by

y = − 25 x − 2

3

7. Find an equation for the line that

a) is parallel to the line defined by y = 2x + 3 and passes through the point (4, 5)

b) is parallel to the line defined by y = 32 x + 1

and passes through the point (−2, 3) c) is perpendicular to the line defined by y = −3x + 2 and passes through the point (1, −5)

d) is perpendicular to the line defined by

y = − 34 x − 5 and passes through the point

(−2, − 4)

Similar and Congruent Triangles 8. ABC is similar to PQR.

a) Find the measure of ∠P. b) Find the length of QR.

Name: ___________________________________ Date: _______________________________

…BLM 2–3... (page 1)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 2–3 Section 2.1 Practice Master

Section 2.1 Practice Master

1. Determine the coordinates of the midpoint of each line segment. a) b) c)

2. Determine the coordinates of the midpoint of the line segment defined by each pair of endpoints. a) (1, 5) and (7, 3) b) (− 4, −3) and (5, 2) c) (−3.2, 4.1) and (5.6, −2.3)

d) ( )2 3,5 4− − and ( )4 3,5 4

3. Find the slope of each median shown.

a) b)

4. The endpoints of the diameter of a circle are

A(−5, −3) and B(3, 7). Find the coordinates of the centre of this circle.

Name: ___________________________________ Date: _______________________________

…BLM 2–3... (page 2)


5. The vertices of PQR are P(−3, 5), Q(5, 7), and R(3, −3). a) Find an equation in slope y-intercept form

for the median from vertex P. b) Find an equation in slope y-intercept form

for the median from vertex Q. c) Find an equation in slope y-intercept form

for the median from vertex R. 6. One endpoint of a diameter of a circle

centred at the origin is (−5, 2). Find the coordinates of the other endpoint of this diameter.

7. Determine an equation for the right bisector

of the line segment with endpoints D(−3, 5) and M(7, −9).

8. a) Draw JKL with vertices J(−6, 4),

K(−4, −5), and L(6, 1). b) Draw the median from vertex J. Then,

find an equation in slope y-intercept form for this median.

c) Draw the right bisector of KL. Then, find an equation in slope y-intercept form for this right bisector.

9. Write an expression for the coordinates of the midpoint of the line segment with endpoints A(2a, 3b) and B(4a, 5b). Explain your reasoning.

10. a) Draw ABC with vertices A(−8, 0),

B(0, 0), and C(0, −8). b) Construct the midpoints of AB, BC,

and AC and label them D, E, and F, respectively.

c) Join the midpoints to form DEF. d) Show that the length of line segment DE

is one half the length of line segment AC. e) Show that the length of line segment DF

is one half the length of line segment BC. f) Show that the length of line segment EF

is one half the length of line segment AB.

Name: ___________________________________ Date: _______________________________

…BLM 2–4...



1. Estimate the length of each line segment from its graph. Then, calculate its exact length. a)

b)

2. Calculate the length of the line segment defined by each pair of endpoints. a) (−5, 6) and (3, −2) b) (−7, −5) and (− 4, 6) c) (− 4.7, 3.8) and (6.4, −5.4)

d) ( )3 2,4 5− − and ( )1 3,4 5

3. A circle has a diameter with endpoints

R(− 4, 6) and T(10, −8). a) Find the length of this diameter. b) Find the length of the radius of this circle.

4. For the triangle with vertices A(−3, 5),

B(5, 3), and C(1, −5), determine the length of a) the median from A b) the median from B c) the median from C

5. The vertices of XYZ are X(−6, 8), Y(−2, − 4), and Z(4, 6). a) Determine the exact length of each side

of this triangle. b) Classify the triangle. c) Determine the perimeter of the triangle.

Round your answer to the nearest tenth of a unit.

6. a) Show that the triangle with vertices

D(−3, 0), E(0, 4), and F(3, 0) is isosceles. b) List the coordinates of another isosceles

triangle. 7. a) Determine the length of the median from

vertex A in the triangle with vertices A(−6, 5), B(−2, 8), and C(4, − 4).

b) Describe how you could use geometry software to verify your answer to part a).

8. a) Determine the area of the right triangle

with vertices D(−3, −1), E(3, 2), and F(1, 6).

b) Describe how you could use geometry software to verify your answer to part a).

9. A line segment has endpoints S(− 4, −5) and

T(10, 7). a) Find the coordinates of the midpoint of line

segment ST. b) Verify your answer to part a) by

determining the distance from the midpoint to each of the endpoints and the distance between the endpoints.

Name: ___________________________________ Date: _______________________________

…BLM 2–6...



1. Find an equation for the line containing line segment CD.

2. A triangle has vertices D(1, 3), E(4, 1), and

F(6, 4). a) Draw DEF. b) Use analytic geometry to verify that

∠DEF is a right angle. 3. Find the length of the median from vertex S.

4. A quadrilateral has vertices P(−1, 3), Q(5, 4),

R(4, −2), and S(−2, −3). a) What type of quadrilateral is PQRS?

Explain. b) Determine the perimeter of PQRS.

Round your answer to the nearest tenth of a unit. 5. The endpoint of a radius of a circle with

centre C(2, 3) is D(5, 5). Determine a) the length of the radius of the circle b) the coordinates of the endpoint E of the

diameter DE of the circle

6. In ABC, D is the midpoint of AB and E is the midpoint of AC.

a) Find the coordinates of D and E. b) Show that DE is parallel to BC. c) Show that DE is half the length of BC.

7. The coordinates of the vertices of a triangle

are D(−5, 2), E(2, 5), and F(2, −1). a) Draw DEF. b) Classify DEF.

8. Determine the shortest distance from

a) the point (6, 3) to the line y = −2x + 1

b) the point (−5, 3) to the line y = 23 x + 2

c) the point (4, −5) to the line joining C(−3, 1) and D(6, 4) 9. The points W(−2, −2), X(−6, 2), and Y(2, 5)

are three vertices of parallelogram WXYZ. a) Find the coordinates of vertex Z. b) Find the length of the diagonals XZ

and WY. c) Show that the diagonals XZ and WY

bisect each other. 10. A triangle has vertices A(−4, 2), B(−2, −6),

and C(6, −2). a) Determine the length of the median from

vertex A. b) Determine an equation in the form

y = mx + b for the median from vertex A.

Name: ___________________________________ Date: _______________________________

…BLM 2–8...



1. Determine an equation for each circle. a) b)

2. State the radius of the circle defined by each equation and give the coordinates of one point on the circle. a) x2 + y2 = 49 b) x2 + y2 = 16 c) x2 + y2 = 64 d) x2 + y2 = 1.44

3. Find an equation for the circle centred at the

origin that passes through each point. a) (3, − 4) b) (−5, 2) c) (3, 7) d) (−6, −2)

4. Determine whether each point is on, inside,

or outside the circle defined by x2 + y2 = 26. a) (1, 3) b) (− 4, 6) c) (1, 5)

5. Determine an equation for the circle that has a diameter with endpoints B(− 4, 7) and C(4, −7).

6. The point A(4, b) lies on the circle defined by x2 + y2 = 25. a) Find the possible value(s) of b. b) Use a graph to show that the point(s)

corresponding to the possible value(s) of b are on the circle.

7. a) Graph the circle defined by x2 + y2 = 13.

b) Verify algebraically that the points M(−3, 2) and N(2, −3) are on the circle.

c) Find an equation in the form y = mx + b for the right bisector of chord MN.


b) Verify algebraically that the line segment joining P(−3, 6) and Q(6, −3) is a chord of this circle.

c) Find an equation in the form y = mx + b for the right bisector of chord PQ.


b) Verify algebraically that the point D(6, −8) lies on this circle.

c) Construct the line segment DO. Determine the slope of the radius DO.

d) Draw the line that is perpendicular to the line segment DO through the point D. Determine the slope of this line.

e) Determine an equation for the tangent line in part d).

Name: ___________________________________ Date: _______________________________

…BLM 2–9... (page 1)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 2–9 Chapter 2 Review

Chapter 2 Review

2.1 Midpoint of a Line Segment 1. Find the midpoint of each line segment.

a) b)

2. a) Determine the midpoint of the line segment with endpoints E(−6, 7) and F(−2, 1). b) Determine the midpoint of the line

segment with endpoints E(−5, −9) and F(2, 4). 3. a) Draw the triangle with vertices A(−5, 2),

B(−1, − 4), and C(3, 3). b) Draw the median from vertex A. Then,

find an equation in the form y = mx + b for this median.

c) Draw the right bisector of AC. Then, find an equation in the form y = mx + b for this right bisector.

d) Draw the altitude from vertex C. Then, find an equation in the form y = mx + b for this altitude.

2.2 Length of a Line Segment 4. Determine the length of the line segment

defined by each pair of points. a) R(−5, 6) and S(−2, 6) b) T(4, −5) and U(4, 5) c) M(−5, 6) and N(3, −4) d) P(−2, 6) and Q(7, −3)

5. a) Determine the length of the median from

vertex R of PQR. b) Determine the perimeter of PQR.


6. a) Draw the triangle with vertices X(1, 4), Y(−3, −2), and Z(3, −6). b) Use analytic geometry to show that

∠XYZ = 90°. c) Determine the area of XYZ.

2.3 Apply Slope, Midpoint, and Length Formulas 7. Show that the triangle with vertices P(−1, 0),

Q(0, 3 ), and R(1, 0) is equilateral.

Name: ___________________________________ Date: _______________________________

…BLM 2–9... (page 2)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 2–9 Chapter 2 Review

8. a) Show algebraically that this triangle is isosceles.

b) Find the midpoints of the equal sides. c) Show algebraically that the line segment

joining the midpoints of the equal sides is parallel to the third side of the triangle. 9. On a map, a ski hill has a chair lift running

straight from A(30, 25) to B(60, 55). a) How long is the section of the chair lift if

each unit on the map grid represents 1 m, to the nearest tenth of a metre?

b) Is the point C(50, 45) on the chair lift? Explain your reasoning.

2.4 Equation for a Circle 10. Determine an equation for each circle. a)

b) c)

11. Find an equation for the circle that is centred at the origin and a) has a radius of 3.7 b) has a radius of 8 c) has a diameter of 18 d) passes through the point (3, 5)

12. a) Show that the line segment joining

C(−2, 5) and D(−5, 2) is a chord of the circle defined by x2 + y2 = 29. b) Determine an equation for the right

bisector of the chord CD. 13. a) Show that point B(−3, −2) lies on the

circle defined by x2 + y2 = 13. b) Find an equation for the radius from the

origin O to point B. c) Find an equation for the line that passes

through B and is perpendicular to OB.

Name: ___________________________________ Date: _______________________________

…BLM 2–11...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 2–11 Chapter 2 Practice Test

Chapter 2 Practice Test

1. The midpoint of the line segment with endpoints A(− 4, −5) and B(2, 3) is A (−3, −4) B (−2, −2) C (−1, −1) D (−4.5, 2.5)

2. The length of the line segment with endpoints C(−3, −5) and D(2, − 4) is A 82 B 40 C 106 D 26

3. An equation for the circle with centre (0, 0) and radius 8 is A x2 + y2 = 64 B x2 + y2 = 16 C x2 + y2 = 8 D x2 + y2 = 2

4. The endpoints of a diameter of a circle are A(−3, 7) and B(5, −3). The coordinates of the centre of this circle are A (− 4, 5) B (1, 2) C (13, −13) D (−11, 17)

5. The point (−4, 5) lies on a circle with centre (0, 0). An equation for the circle is A x2 + y2 = 20 B x2 + y2 = 9 C x2 + y2 = 1 D x2 + y2 = 41

6. Find the midpoint and the length of the line segment defined by each pair of endpoints. a) A(−9, −2) and B(5, −4) b) C(−2, −5) and D(5, −2)

7. a) Draw the triangle with vertices A(−5, −2), B(−1, 6), and C(3, −1). b) Determine an equation for the median

from A. c) Determine an equation for the

perpendicular bisector of AB.

8. The library is located exactly halfway between Brandon’s house and Vaughn’s house. The intervals on the grid represent 1 km.

a) How far apart are Brandon’s house and

Vaughn’s house, to the nearest tenth of a kilometre?

b) Determine the coordinates of the library.

9. The vertices of a triangle are D(−4, −2), E(−2, 6), and F(6, −4). a) Determine the lengths of the sides of the

triangle. b) Classify DEF. Explain your reasoning. c) Determine the perimeter of DEF.


d) Describe how you could use geometry software to verify your answers in parts a), b), and c).

10. a) Plot the triangle with vertices G(−5, −4), H(−1, 8), and I(3, −6). b) Determine an equation for the median

from vertex G. c) Determine an equation for the right

bisector of GH. d) Determine an equation for the altitude

from G to HI.

Name: ___________________________________ Date: _______________________________

…BLM 2–12...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 2–12 Chapter 2 Test

Chapter 2 Test

1. The midpoint of the line segment with endpoints A(−3, 5) and B(7, −1) is A (2, 2) B (−5, 3) C (4, 4) D (1, 3)

2. The length of the line segment with endpoints C(−5, 2) and D(3, −1) is A 17 B 13 C 65 D 73

3. An equation for the circle with centre (0, 0) and radius 6 is A x2 + y2 = 6 B x2 + y2 = 9 C x2 + y2 = 36 D x2 + y2 = 12

4. The endpoints of a diameter of a circle are A(−4, 3) and B(2, −5). The coordinates of the centre of this circle are A (−3, −4) B (−1, −1) C (3, 4) D (−1, −3)

5. The point (6, −3) lies on a circle with centre (0, 0). The equation of the circle is A x2 + y2 = 45 B x2 + y2 = 3 C x2 + y2 = 9 D x2 + y2 = 81

6. Find the midpoint and the length of the line segment defined by each pair of endpoints. a) A(−5, 4) and B(3, −6) b) C(−4, −3) and D(1, −2)

7. a) Draw the triangle with vertices A(−5, −3), B(1, 5), and C(3, −1). b) Determine an equation in slope

y-intercept form for the median from B. c) Determine an equation for the

perpendicular bisector of AB.

8. A computer store is located exactly halfway between David’s house and his school. The intervals on the gridlines represent 1 km.

a) How far apart are David’s house and

his school, to the nearest tenth of a kilometre?

b) Determine the coordinates of the computer store.

9. The vertices of a triangle are D(−4, 5), E(−7, −1), and F(−1, −1). a) Determine the lengths of the sides of

the triangle. b) Classify DEF. Explain your reasoning. c) Determine the perimeter of DEF.


d) Describe how you could use geometry software to verify your answers to parts a), b), and c).

10. a) Plot the triangle with vertices X(−4, −3), Y(−2, 5), and Z(4, 1). b) Determine an equation in slope

y-intercept form for the median from vertex X.

c) Determine an equation for the right bisector of XY.

d) Determine an equation for the altitude from Y to XZ.

…BLM 2–14... (page 1)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited Chapter 2 Practice Masters Answers

BLM Answers

Get Ready 1. a) x = 2 b) y = 6 c) z = − 40 d) d = 25 2. a) y = x + 3 b) y = −5x + 7

c) 1 42 3y x= − + d) 1 2

15 5y x= +

3. a) 2 b) −1

c) −8 d) 53−

4. a) y = 5x − 17 b) 1 173 3y x= +

c) y = − 4x + 7 d) 1 34y x= − −

5. a) y = 2x + 1 b) y = −x + 2 c) y = −3x − 13 d) y = 3x − 1

6. a) m = 5 b) m = 14

c) m = 13− d) m = 5

2

7. a) y = 2x − 3 b) 3 62y x= +

c) 1 163 3y x= − d) 4 4

3 3y x= −

8. a) 60° b) 5 cm


1. a) ( )1 , 22 b) (1, 1) c) (−1, −1)

2. a) (4, 4) b) ( )1 1,2 2−

c) (1.2, 0.9) d) ( )1 , 05

3. a) 65m = − b) 9

11m =

4. (−1, 2)

5. a) 3 267 7y x= − + b) 6 15y x= +

c) 9 212 2y x= − +

6. (5, −2)

7. 5 247 7y x= −

8. a), b), c)

9. (3a, 4b); Use the midpoint formula with x1 = 2a,

x2 = 4a, y1 = 3b, and y2 = 5b. 10. a), b), c)

d) From the graph, DE is 4 units long and AC is 8 units

long, so DE is one half the length of AC. e) From the graph, DF is 4 units long and BC is 8 units

long, so DF is one half the length of BC. f ) By the Pythagorean theorem,

2 2 2 2 2 2

2 2 2 2

EF DF DE AB AC BC

4 4 8 832 128

EF 32 AB 128

16 2 64 2

16 2 64 2

4 2 8 2

= + = +

= + = += =

= =

= × = ×

= × = ×

= =

Line segment AB is twice the length of line segment EF.

…BLM 2–14... (page 2)


Section 2.2 Practice Master 1. Estimates may vary. Exact answers: a) 29 b) 58 2. a) 128 b) 130 c) 207.85 d) 2 3. a) 392 b) 98 4. a) 72 b) 45 c) 9 5. a) XY = 160 , XZ = 104 , YZ = 136 b) scalene c) 34.5 6. a) DE = 5, EF = 5, DF = 6; DE = EF,

so DEF is isosceles. b) Answers may vary. For example: A(−5, 0), B(0, 10),

C(5, 0) 7. a) 58 b) Construct ABC by plotting the vertices and

connecting them with line segments. Construct the midpoint, D, of side BC. Construct line segment AD. Select line segment AD and measure its length.

8. a) 15 square units b) Construct DEF by plotting the vertices and

connecting them with line segments. Select the vertices and then construct the triangle interior. Select the triangle interior and measure its area.

9. a) M(3, 1) b) SM = 85 , MT = 85

ST 340

4 85

4 85

2 852 SM2 MT

=

= ×

= ×

== ×= ×

So, ST = 2 × SM = 2 × MT. M is the midpoint of ST.

Section 2.3 Practice Master 1. y = −2x − 1 2. a)

b) slope DE = 23− ; slope EF = 3

2 ; since the slopes

are negative reciprocals, DE is perpendicular to EF, and ∠DEF is a right angle.

3. 65 4. a) PQRS is a rhombus, because all four sides are equal

in length. b) 24.3 units

5. a) 13 b) E(−1, 1) 6. a) D(−1, 5), E(−3, 1) b) slope DE = 2; slope BC = 2; since the slopes are

equal, the line segments are parallel. c) DE = 20

BC 80

4 20

4 20

2 20

=

= ×

= ×

=

Thus, DE is half the length of BC. 7. a) b) DEF is isosceles. 8. a) 39.2 b) 13 c) 62.5 9. a) Z(6, 1) b) XZ = 145 , WY = 65 c) The midpoint of XZ and the midpoint of WY both

occur at (0, 1.5). Thus, XZ and WY bisect each other.

10. a) 72 b) y = −x − 2

Section 2.4 Practice Master 1. a) x2 + y2 = 36 b) x2 + y2 = 7 2. Points may vary. Examples are given. a) 7; (0, 7) b) 4; (4, 0) c) 8; (−8, 0) d) 1.2; (0, −1.2) 3. a) x2 + y2 = 25 b) x2 + y2 = 29 c) x2 + y2 = 58 d) x2 + y2 = 40 4. a) inside b) outside c) on 5. x2 + y2 = 65 6. a) 3, −3 b)

…BLM 2–14... (page 3)


7. a) b) M(−3, 2):

2 2

2 2

13

( 3) 29 413

x y= + =

= − += +=

L.S. R.S.

L.S. = R.S. N(2, −3):

2 2

2 2

13

2 ( 3)4 913

x y= + =

= + −= +=

L.S. R.S.

L.S. = R.S. c) y = x 8. a)

b) Check that both endpoints are on the circle. P(−3, 6):

2 2

2 2

45

( 3) 69 3645

x y= + =

= − += +=

L.S. R.S.

L.S. = R.S. Q(6, −3):

2 2

2 2

45

6 ( 3)36 945

x y == +

= + −= +=

R.S.L.S.

L.S. = R.S. c) y = x 9. a), c), d)

b) D(6, −8)

2 2

2 2

100

6 ( 8)36 64100

x y= + =

= + −= +=

L.S. R.S.

L.S. = R.S.

c) 43− d) 3

4 e) 3 254 2y x= −

Chapter 2 Review

1. a) ( )11, 2 b) ( )1 1,2 2−

2. a) (− 4, 4) b) ( )3 5,2 2− −

…BLM 2–14... (page 4)


3. a), b), c), d)

b) 5 112 12y x= − − c) 118 2y x= − −

d) 2 13y x= +

4. a) 3 b) 10 c) 164 d) 162 5. a) 53 b) 21.8 6. a)

b) slope XY = 32 ; slope YZ = 2

3− ; since the slopes

are negative reciprocals, ∠XYZ = 90°. c) 26 square units 7. PQ = QR = PR = 2; all three sides have equal length,

so PQR is equilateral. 8. a) DE = EF = 26 ; since two sides have equal

length, DEF is isosceles.

b) ( )3 1G ,2 2− is the midpoint of DE; ( )1 7H ,2 2 is

the midpoint of EF.

c) slope GH = 32 ; slope DF = 3

2 ; since the slopes are

equal, the two segments are parallel. 9. a) 42.4 m b) Yes. The line y = x − 5 contains the points A and B.

Since it also contains the point C, and C is between A and B, C is on the chair lift.

10. a) x2 + y2 = 49 b) x2 + y2 = 37 c) x2 + y2 = 6.25

11. a) x2 + y2 = 13.69 b) x2 + y2 = 8 c) x2 + y2 = 81 d) x2 + y2 = 34 12. a) Check that both endpoints are on the circle. C(−2, 5):

2 2

2 2

29

( 2) 54 2529

x y= + =

= − += +=

L.S. R.S.

L.S. = R.S. D(−5, 2):

2 2

2 2

29

( 5) 225 429

x y= + =

= − += +=

L.S. R.S.

L.S. = R.S. b) y = −x 13. a) Check that the point B(–3, –2) satisfies the equation

x2 + y2 = 13. 2 2

2 2

13

( 3) ( 2)9 413

x y == += − + −= +=

=

R.S.L.S.

L.S. R.S.

b) 23y x= c) 3 13

2 2y x= − −

Chapter 2 Practice Test 1. C 2. D 3. A 4. B 5. D

6. a) (−2, −3); 200 b) ( )3 7,2 2− ; 58

7. a)

b) 3 74 4y x= + c) 1 1

2 2y x= − +

8. a) 7.2 km b) (5, 5)

…BLM 2–14... (page 5)


9. a) DE = 68 ; EF = 164 ; DF = 104 b) DEF is scalene because the three sides have

different lengths. c) 31.3 units d) Construct DEF by plotting the vertices and

connecting them with line segments. Select the sides of the triangle and then measure their lengths. Select the vertices of the triangle and then construct the triangle interior. Select the triangle interior and measure the perimeter.

10. a)

b) 5 16 6y x= + c) 1 13y x= − + d) 2 18

7 7y x= −

Chapter 2 Test 1. A 2. D 3. C 4. B 5. A 6. a) (−1, −1); 164

b) ( )3 5,2 2− − ; 26

7. a)

b) 7 32 2y x= + c) 3 1

4 2y x= − −

8. a) 10.2 km b) (6, 5) 9. a) DE = 45 ; EF = 6; DF = 45 b) isosceles; DE = DF c) 19.4 units d) Construct DEF by plotting the vertices and

connecting them with line segments. Select the sides of the triangle and then measure their lengths. Select the vertices of the triangle and then construct the triangle interior. Select the triangle interior and measure the perimeter.

10. a)

b) 6 95 5y x= + c) 1 1

4 4y x= − + d) y = −2x + 1

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