1. Given Triangle ABC with vertices A(0,0), B(4,8), and C(6,2).

a) Show that Triangle ABC is an isosceles triangle.

AB222 cba 222 48 c21664 c

280 c 280 cc80

BC222 26 c2436 c

240 c 240 cc40

AC222 62 c2364 c

240 c 240 cc40

Since two legs of the triangle are congruent, triangle ABC is isosceles.

1. Given Triangle ABC with vertices A(0,0), B(4,8), and C(6,2).

b) Find the coordinates of D, the midpoint of the base.

Midpoint: (2,4)

1. Given Triangle ABC with vertices A(0,0), B(4,8), and C(6,2).

c) Show that CD is perpendicular to AB.

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24

8m

AB CD

2

1

4

2

m

Since the slopes of AB and CD are negative reciprocals, AB is perpendicular to CD.

D

6a. Given the verticies of PQRS with P(0,0), Q(4,3), R(7,-1), and S(3,-4) Show that PQRS is a rhombus.

PQ222 cba 222 43 c2169 c

225 c 225 cc5

QR222 43 c2169 c

225 c 225 cc5

RS222 43 c2169 c

225 c 225 cc5

SP222 43 c2169 c

225 c 225 c

c5

Since all four sides of quadrilateral PQRS are congruent, PQRS is a rhombus.

SPRSQRPQ

6b. Show that PQRS is a square.

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4

3m

PQ

3

4

3

4

m

QR

Since each pair of consecutive sides of the rhombus have slopes that are negative reciprocals, they are all perpendicular and form four right angles.

4

3m

RS

3

4

3

4

m

SP

Since PQRS is a rhombus with all right angles, it is a square.

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2. Given Triangle RST with vertices R(0,6), S(2,0), and T(8,2).

a.) Show that Triangle RST is a right triangle and state a reason for your conclusion.

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32

6

m

RS ST

3

1

6

2m

The slopes of two consecutive sides are negative reciprocals, which makes them perpendicular, forming a right angle.

Therefore, triangle RST is a right triangle.

3. Given triangle PQR with vertices P(2,4), Q(5,8), and R(9,5).

a.) Show that Triangle PQR is isosceles.

b.) Show that Triangle PQR is a right triangle.

PQ222 cba 222 43 c2169 c

225 c 225 cc5

QR222 43 c2169 c

225 c 225 cc5

PR222 71 c2491 c

250 c 250 cc50

Since two legs of the triangle are congruent, PQR is an isosceles triangle.

QRPQ

3. Given triangle PQR with vertices P(2,4), Q(5,8), and R(9,5).

a.) Show that Triangle PQR is isosceles.

b.) Show that Triangle PQR is a right triangle.

5PQ

5QR

50PR

222 5055

502525

5050

Since the lengths of the sides of the triangles work in the pythagorean theorem, triangle PQR is a right triangle.

?

?

7a) The vertices of quadrilateral ABCD are A(1,1), B(3,4), C(9,1) and D(7,-2). Show that the quadrilateral is a parallelogram.

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2

3m

AB BC

2

1

6

3

m

CD

2

3m

DA

2

1

6

3

m

Since the slopes of both pairs of opposite sides are equal, they are parallel.

Quadrilateral ABCD, with both pairs of opposite sides parallel, is a parallelogram.

KN

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22

4m

KN222 cba 222 24 c2416 c

220 c 220 cc20

LM

22

4m

LM

Since one pair of opposite sides of quadrilateral KLMN are both parallel and congruent, KLMN is a parallelogram.

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222 24 c2416 c

220 c 220 cc20

Since the slopes of KN and LM are equal, KN is parallel to LM.

LMKN

13) Quadrilateral DEFG has vertices D(-4,0), E(0,1), F(4,-1) and G(-4,-3).

A) show that DEFG is a trapezoid.

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4

1m

DE EF

2

1

4

2

m

FG

4

1

8

2m

GD

0

3m

Therefore quadrilateral DEFG with only one pair of opposite sides parallel is a trapezoid.

undefined

The slopes of DE and FG are equal, so DE is parallel to FG. The slopes of EF and GD are not equal, so EF is not parallel to GD.

13) Quadrilateral DEFG has vertices D(-4,0), E(0,1), F(4,-1) and G(-4,-3).

B) Show that DEFG is not an isosceles trapezoid.

222 cba 3

DG EF

Since the lengths of the nonparallel sides are not equal, DEFG is not an isosceles trapezoid.

222 42 c2164 c220 c 220 cc20

13) Quadrilateral DEFG has vertices D(-4,0), E(0,1), F(4,-1) and G(-4,-3).

C) Show that diagonals DF and EG do not bisect each other.

DF

Since the midpoints of the two diagonals are not the same point, diagonals DF and EG do not bisect each other.

221 xx

xm

2

21 yyym

2

44mx

02

0mx

2

10 my

2

1my

EG

2

04mx

22

4

mx

2

31 my

12

2

my

2

1,0M

1,2 M

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22

4m

AB BC

3

1

6

2m

22

4m

CD DA

3

1

6

2m

Since the slopes of both pairs of opposite sides are equal, the opposite sides are parallel.

9. Quadrilateral ABCD has vertices A(4,4), B(2,0), C(-4,-2) and D(-2,2). Show that ABCD is a parallelogram.

Therefore quadrilateral ABCD, with both pairs of opposite sides parallel, is a parallelogram.

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14) A(1,3), B(-1,1), C(-1,-2) and D(4,3) are the vertices of quadrilateral ABCD. Show that ABCD is an isosceles trapezoid.

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12

2m

AB BC

3

0m

CD

15

5m

DA

0

3m

undefined

SLOPE

Since the slopes of AB and CD are equal, AB is parallel to CD. The slopes of BC and DA are not equal, so BC is not parallel to DA.

Quadrilateral ABCD with only one pair of parallel sides is a trapezoid.

14) A(1,3), B(-1,1), C(-1,-2) and D(4,3) are the vertices of quadrilateral ABCD. Show that ABCD is an isosceles trapezoid.

3BC 3AD

Distance

Since the lengths of the nonparallel sides of trapezoid ABCD are congruent, ABCD is an isosceles trapezoid.

ADBC

15. The vertices of triangle ABC are A(0,10), B(5,0), and C(8,4)

a.) Show that triangle ABC is a right triangle.

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4

3

8

6

m

AC BC

3

4m

Therefore, triangle ABC is a right triangle.

The slopes of two consecutive sides are negative reciprocals, which makes them perpendicular, forming a right angle.

15. The vertices of triangle ABC are A(0,10), B(5,0), and C(8,4)

b.) Point D(4,2) lies on AB. Show that CD is the altitude to AB.

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25

10

m

AB CD

2

1

4

2m

D

The slopes of AB and CD are negative reciprocals, which makes AB perpendicular to CD.

Since CD extends from a vertex perpendicular to the base, it is the altitude to AB.

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2

3

4

6m

PQ QR

3

2m

RS SP

10. Quadrilateral PQRS has vertices P(0,2), Q(4,8), R(7,6) and S(3,0). Show that ABCD is a rectangle.

2

3

4

6m

3

2m

The slopes of all of the consecutive sides are negative reciprocals, therefore they are perpendicular, forming right angles.

Therefore, quadrilateral PQRS with 4 right angles is a rectangle.

16. A(0,2), B(9,14), and C(12,2) are the vertices of triangle ABC. D(6,10) is a point on AB and E(8,2) is a point on AC.

a.) Show that DE is parallel to BC.

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42

8

m

DE BC

43

12

m

Since the slopes of DE and BC are equal, DE is parallel to BC.

12. Draw any quadrilateral. Find the midpoints of each side of your quadrilateral. Show that the quadrilateral formed from connecting the midpoints is always a parallelogram.

HOMEWORK

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