Investigation 4Dilations and Similar

FiguresProblem 4.1: Focus on Dilations

Part ADraw the image of quadrilateral PQRS after a dilation with center (0, 0) and scale factor 3 on the graph to the right. Label the corresponding points P’, Q’, R’, and S’.

How do the side lengths of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?

Lengths of the sides of PQRS:PQ =

PS =

QR =

RS =

Lengths of the sides of P’Q’R’S’:P’Q’ =

P’S’ =

Q’R’ =

R’S’ =

The side lengths of quadrilateral P’Q’R’S’ are 3 times as long as the side lengths of PQRS.

How do the angle measures of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?

m<Q =

m<P =

m< S =

m <R =

m<Q’ =

m<P’ =

m< S’ =

m<R’ =

Corresponding angles have equal measure.

How does the perimeter (distance around the shape) of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?

Quadrilateral PQRSPQ = 2 unitsPS = 2 unitsSR = unitsQR = 3 units

Perimeter of PQRS2 + 2 + + 3 = 7 + units

Quadrilateral P’Q’R’S’P’Q’ = 6 unitsP’S’ = 6 unitsS’R’ = 3 unitsQ’R’ = 9 units

Perimeter of P’Q’R’S’6 + 6 + 3 + 9 = 21 + 3 units

The perimeter of P’Q’R’S’ is 3 times the perimeter of PQRS.

How does the area of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?

Area Formula for a Trapezoid

A = ½ h (b1 + b2)

Area of PQRSA = ½ h (b1 + b2)

A = ½ (2)(3 + 2)

A = 5 square units

Area of P’Q’R’S’A = ½ h (b1 + b2)

A = ½ (6) (9 + 6)

A = 45 square unitsThe area of P’Q’R’S’ 9 times as large as the area of PQRS.

How do the slopes of the sides of quadrilateral P’Q’R’S’ compare to the slopes of the sides of quadrilateral PQRS?

Slopes for PQRS:

PQ =

PS =

SR =

QR =

Slopes for P’Q’R’S’:

P’Q’ =

P’S’ =

S’R’=

Q’R’ =

Corresponding sides have the same slope.

Point P Q R S

Original Coordinates (2, 4) (2, 2) (5, 2) (4, 4)

Coordinates after a dilation with center (0, 0) and scale factor of 3

(6, 12) (6, 6) (15, 6) (12, 12)

What rule of the form (x, y) (_____ , _____) shows how coordinates of corresponding points are related under a dilation with center (0, 0) and scale factor of 3?

Rule for how coordinates of corresponding points are related under a dilation with center (0, 0) and scale factor of 3:

(x, y) (3x, 3y)

Part CDraw the image of quadrilateral PQRS after a dilation with center (0, 0) and scale factor ½.

How do the side lengths of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?

How do the side lengths of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?Lengths of the sides of PQRS:

PQ =

PS =

QR =

RS =

Lengths of the sides of P’Q’R’S’:P’Q’ = Q’R’ =

P’S’ = R’S’ =

The side lengths of quadrilateral P’Q’R’S’ are ½ times as long as the side lengths of PQRS.

How do the angle measures of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?

m<Q =

m<P =

m< S =

m<R =

Corresponding angles have equal measure.

m<Q’ =

m<P’ =

m< S’ =

m <R’ =

How does the perimeter of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS? Quadrilateral PQRS

PQ = 2 unitsPS = 2unitsSR = 2unitsQR = 8 units

Perimeter of PQRS2 + 2 + + 8 =

8 + 4 + 2 units

Quadrilateral P’Q’R’S’P’Q’ = unitsP’S’ = unitsS’R’ = unitsQ’R’ = 4 units

Perimeter of P’Q’R’S’ + + + 4 =

4 + 2 + units

The perimeter of P’Q’R’S’ is ½ times the perimeter of PQRS.

How does the area of quadrilateral P’Q’R’S’ compare to those of quadrilateral PQRS?Area of PQRS Area of P’Q’R’S’

The area of P’Q’R’S’ times as large as the area of PQRS.

How do the slopes of the sides of quadrilateral P’Q’R’S’ compare to the slopes of the sides of quadrilateral PQRS?

Slopes for PQRS:

PQ =

PS =

SR =

QR =

Slopes for P’Q’R’S’:

P’Q’ =

P’S’ =

S’R’=

Q’R’ =

Corresponding sides have the same slope.

What rule of the form (x, y) (_____ , _____) shows how coordinates of corresponding points are related under a dilation with center (0, 0) and scale factor of ½?

Point P Q R S

Original Coordinates (-2, 4) (-4, -4) (4, -4) (6, 2)

Coordinates after a dilation with center (0, 0) and scale factor of ½

(-1, 2) (-2, -2) (2, -2) (3, 1)

Rule for how coordinates of corresponding points are related under a dilation with center (0, 0) and scale factor of ½:

(x, y) (½x, ½y)