©thevisualclassroom.com

2.11 Verifying Geometric Properties

When proving sides of a quadrilateral are parallel use slope 2 1

2 1

y ym

x x

When proving sides are perpendicular, use slope

When proving sides are equal in length, use distance

2 22 1 2 1( ) ( )d x x y y

©thevisualclassroom.com

Proving two line segments bisect each other, use midpoint

1 2 1 2,2 2

x x y y

©thevisualclassroom.com

Ex. 1: Given rectangle A(–2, 0), B(–3, 4), C(5, 6), D(6, 2). Prove that the diagonals of rectangle ABCD are equal in length. Find the lengths of AC and BD

A(–2, 0)

B(–3, 4)C(5, 6)

D(6, 2)

2 22 1 2 1( ) ( )d x x y y

2 2(5 2) (6 0)ACd

49 36ACd

85ACd

2 2( 3 6) (4 2)BDd

81 4BDd

85BDd

©thevisualclassroom.com

Ex. 2: Given rectangle A(–2, 0), B(–3, 4), C(5, 6), D(6, 2). Prove that the diagonals bisect each other.Find the midpoints of AC and BD

A(–2, 0)

B(–3, 4)C(5, 6)

D(6, 2)

2 5 0 6,

2 2ACM

1.5,3ACM

3 6 4 2,

2 2BDM

1.5,3BDM

1 2 1 2,2 2

x x y y

©thevisualclassroom.com

Ex. 3: Prove the midsegments of P (–4, 5), Q(6, 7), R(4,–3), S(–6,–1) form a parallelogram.Find the midpoints of the sides

P(–4, 5)

Q(6, 7)

R(4,–3)S(–6,–1)

4 6 5 7,

2 2PQM

1 2 1 2,2 2

x x y y

6 4 7 3,

2 2QRM

4 6 3 1,

2 2RSM

6 4 1 5,

2 2SPM

T(1, 6)T(1,6)

U(5, 2)

U(5,2)

V(–1,–2)

V(–1,–2)

W(–5, 2)

W(–5,2)

©thevisualclassroom.com

P(–4, 5)

Q(6, 7)

R(4,–3)S(–6,–1)

T(1, 6)

T(1,6)

U(5, 2)

U(5,2)

V(–1,–2)

V(–1,–2)

W(–5, 2)

W(–5,2)

Find the slopes of TUVW2 1

2 1

y ym

x x

2 6

5 1TUm

2 2

5 1UVm

2 2

5 1VWm

6 2

1 5WTm

2

3

2

3

= – 1

= – 1