8.7

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Coordinate Proof with Quadrilaterals

8.7 Warm-Up

1. Find the distance between the points A(1, –3) and B(–2, 4).

ANSWER

8.7 Warm-Up

2. Determine if the triangles with the given vertices are similar. A(–3, 3), B(–4, 1), C(–2, –1)D(3, 5), E(2, 1), F(4, –3)

ANSWER ABC and DEF are not similar.

8.7 Example 1

Determine if the quadrilaterals with the given vertices are congruent.

O(0, 0), B(1, 3), C(3, 3), D(2, 0);E(4, 0), F(5, 3), G(7, 3), H(6, 0)

SOLUTION

Graph the quadrilaterals. Show that corresponding sides and angles are congruent.

Use the Distance Formula.

OD = BC = EH = FG = 2Since both pairs of opposite sides in each quadrilateral are congruent, OBCD and EFGH are parallelograms.

OB = DC = EF = HG =

8.7 Example 1

So, O and E are corresponding angles, and O E. By substitution, C G.

Similar reasoning can be used to show that B F and D H.

Because all corresponding sides and angles are congruent, OBCD is congruent to EFGH.

Opposite angles in a parallelogram are congruent, so O C and E G. and are parallel, because both have slope 3, and they are cut by transversal .

8.7 Guided Practice

Find all side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.

1. F(–4, 0), G(–3, 3), H(0, 3), J(–2, 0);P(1, 0), Q(2, 3), R(6, 3), S(4, 0)

ANSWER

not congruent

8.7 Guided Practice

Find all side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.

2. A(–2, –2), B(–2, 2), C(2, 2), D(2, –2);O(0, 0), X(0, 4), Y(4, 4), Z(4, 0)

ANSWER AB = BC = CD = DA = OX = XY = YZ = ZO = 4; all angles are right angles; congruent

8.7 Example 2

Determine if the quadrilaterals with the given vertices are similar.

O(0, 0), B(4, 4), C(8, 4), D(4, 0);O(0, 0), E(2, 2), F(4, 2), G(2, 0)

SOLUTION

Graph the quadrilaterals. Find the ratios of corresponding side lengths.

8.7 Example 2

Because OB = CD and BC = DO, OBCD is a parallelogram.Because OE = FG and EF = GO, OEFG is a parallelogram.

Opposite angles in a parallelogram are congruent, so O F and O C. Therefore, C F .

Parallel lines and are cut by transversal , so B and FEO are corresponding angles, and B FEO.

8.7 Example 2

Likewise, and are parallel lines because both have slope 1, and they are cut by transversal , so D and OGF are corresponding angles, and D OGF .

Because corresponding side lengths are proportional and corresponding angles are congruent, OBCD is similar to OEFG.

8.7 Example 3

Show that the glass pane in the center is a rhombus that is not a square.

SOLUTION

Use the Distance Formula. Each side of ABCD has length units. So, the quadrilateral is a rhombus.

The slope of is 3 and the slope of is –3.

Because the product of these slopes is not –1, the segments do not form a right angle. The pane is a rhombus, but it is not a square.

8.7 Guided Practice

3. If you can show two parallelograms have congruent corresponding angles, are the parallelograms similar? Explain.

ANSWER

For the parallelograms to be similar, the lengths of the corresponding sides must also be proportional.

8.7 Guided Practice

4. Explain how you can use the diagonals of quadrilateral ABCD in Example 3 to prove ABCD is a rhombus.

ANSWER

Use slopes of opposite sides to prove that ABCD is a parallelogram. The diagonals are vertical and horizontal segments, so they are perpendicular. By Theorem 8.11, if the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus.

8.7 Example 4

Without introducing any new variables, supply the missing coordinates for K so that OJKL is a parallelogram.

SOLUTION

Choose coordinates so that opposite sides of the quadrilateral are parallel.

must be horizontal to be parallel to , so the y-coordinate of K is c.

8.7 Example 4

The slopes are equal, so . Therefore, b = x – a, and x = a + b.

Point K has coordinates (a + b, c).

To find the x-coordinate of K, write expressions for the slopes of and . Use x for the x-coordinate of K.

8.7 Example 5

Prove that the diagonals of a parallelogram bisect each other.

SOLUTION

STEP 1 Place a parallelogram with coordinates as in Example 4. Draw the diagonals.

8.7 Example 5

STEP 2 Find the midpoints of the diagonals.

The midpoints are the same. So, the diagonals bisect each other.

8.7 Guided Practice

5. Verify that OJ = LK and JK = OL in Example 4.

ANSWER

OK = LK = and JK = OL = a

8.7 Guided Practice

6. Write a coordinate proof that the diagonals of a rectangle are congruent.

ANSWER

Place rectangle OPQR so that it is in the first quadrant, with points O(0, 0), P(0, b), Q(c, b), and R(c, 0). Use the Distance Formula.

So, the diagonals of a rectangle are congruent.

8.7 Lesson Quiz

1. Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.J(1, 1), K(2, 4), L(5, 4), M(4, 1);N(–1, –3), O(0, 0), P(4, 0), Q(3, –3)

ANSWER

JK = LM = NO = PQ =

KL = JM = 3 but OP = NQ = 4; not congruent

8.7 Lesson Quiz

2. Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are similar.R(–2, 3), S(–2, 1), T(–4, 1), U(–4, 3);V(5, 2), W(5, –2), X(1, –2), Y(1, 2)

ANSWER

RS = ST = TU = RU = 2, VW = WX = XY = VY = 4; corresponding sides are proportional and corresponding angles are ; similar

8.7 Lesson Quiz

ANSWER

(d, c)

3. Without introducing any new variables, supply the missing coordinates for G so that EFGH is a rectangle.