4.8

3. Are these triangles congruent?

y

–1

1

x

4.8 Congruence Transformations and Coordinate GeometryBell Thinger

ANSWER 13

ANSWER (9, 5)2. What point is 6 units to the right of (3, 5)?

1. Find the length of AB for A(2, 7) and B(7, –5).

ANSWER Yes, by the SSS Cong. Post. or by the SAS Cong. Post.

4.8

4.8Example 1

Name the type of transformation demonstrated in each picture.

a.

Reflection in a horizontal lineANSWER

4.8Example 1

Name the type of transformation demonstrated in each picture.

Rotation about a pointANSWER

b.

4.8Example 1

Name the type of transformation demonstrated in each picture.

Translation in a straight pathANSWER

c.

4.8Guided Practice

Name the type of transformation shown.

reflection ANSWER

4.8Example 2

Figure ABCD has the vertices A(– 4 , 3), B(–2, 4), C(–1, 1), and D(–3, 1). Sketch ABCD and its image after the translation (x, y) → (x + 5, y –2).

SOLUTION

First draw ABCD. Find the translation of each vertex by adding 5 to its x-coordinate and subtracting 2 from its y-coordinate. Then draw ABCD and its image.

4.8Example 2

(x, y) → (x +5, y – 2)

A(–4, 3) → (1, 1)

B(–2, 4) → (3, 2)

C(–1, 1) → (4, –1)

D(–3, 1) → (2, –1)

4.8Example 3

You are drawing a pattern for a wooden sign. Use a reflection in the x-axis to draw the other half of the pattern.

Woodwork

SOLUTION

Multiply the y-coordinate of each vertex by –1 to find the corresponding vertex in the image.

4.8Example 3

(x, y) → (x, –y)

(–1, 0) → (–1, 0)

(1, 2) → (1, –2)

(5, 0) → (5, 0)

(–1, 2) → (–1, –2)

(1, 4) → (1, –4)

Use the vertices to draw the image. You can check your results by looking to see if each original point and its image are the same distance from the x-axis.

4.8

2. The vertices of ABC are A(1, 2), B(0, 0), and C(4, 0). A translation of ABC results in the image DEF with vertices D(2, 1), E(1, –1), and F(5, –1). Describe the translation in words and in coordinate notation.

ANSWER

Add one to each x-coordinate and subtract one from each y-coordinate, (x, y) → (x +1, y – 1).

Guided Practice

4.8

The endpoints of RS are R(4, 5) and S(1, –3). A reflection of RS results in the image TU , with coordinates T(4, –5) and U(1, 3). Tell which axis RS was reflected in and write the coordinate rule for the reflection.

3.

ANSWER

x-axis, (x, y) → (x, –y)

Guided Practice

4.8Example 4

Graph AB and CD. Tell whether CD is a rotation of AB about the origin. If so, give the angle and direction of rotation.

a. A(–3, 1), B(–1, 3), C(1, 3), D(3, 1)

This is a 90° clockwise rotation.

m AOC = m BOD = 90°

SOLUTION

4.8Example 4

Graph AB and CD. Tell whether CD is a rotation of AB about the origin. If so, give the angle and direction of rotation.

SOLUTION

b. A(0, 1), B(1, 3), C(–1, 1), D(–3, 2)

This is not a rotation.m AOC < m BOD

4.8Example 5

SOLUTION

S You can see that AC = DF = 3, so AC DF.

The vertices of ABC are A(4, 4), B(6, 6), and C(7, 4). The notation (x, y) → (x + 1, y – 3) describes the translation of

ABC to DEF. Show that ABC DEF to verify

that the translation is a congruence transformation.

4.8Example 5

Because ABC DEF, the translation is a congruence transformation.

ANSWER

A Using the slopes, AB DE and AC DF . If you extend AB and DF to form G, the Corresponding Angles Postulate gives you BAC G and G EDF. Then, BAC EDF by the Transitive Property of Congruence.

S Using the Distance Formula, AB = DE = 2 2 so AB DE . So, ABC DEF by the SAS Congruence Postulate.

4.8

yes; 180° counterclockwise

ANSWER

4. Tell whether PQR is a rotation of STR. If so, give the angle and direction of rotation.

Guided Practice

4.8

Show that PQR STR to verify that the transformation is a congruence transformation.

5.

ANSWER

PQ ST, PR SR, by HL, PQR STR so it is a congruence transformation.

Guided Practice

4.8Exit Slip

1. Use coordinate notation to describe the translation 3 units to the left and 1 unit up.

(x, y) (x – 3, y + 1)ANSWER

4.8Exit Slip

ANSWER

2. Use a reflection in the x-axis to draw the other half of the figure

4.8Exit Slip

ANSWER Yes; 180° clockwise or counterclockwise

3. Tell whether XY is a rotation of GH about the origin If the points are X(–6, 2), Y(–4, 3), G(6, –2), and H(4, –3). If so, give the angle and direction of rotation.

4.8

Homework

Pg 288#13-16