Five-Minute Check (over Chapter 8)

Then/Now

New Vocabulary

Key Concept: Reflection in a Line

Example 1: Reflect a Figure in a Line

Example 2: Real-World Example: Minimize Distance by Using a Reflection

Example 3: Reflect a Figure in a Horizontal or Vertical Line

Key Concept: Reflection in the x- or y-axis

Example 4: Reflect a Figure in the x- or y-axis

Key Concept: Reflection in Line y = x

Example 5: Reflect a Figure in the Line y = x

Concept Summary: Reflection in the Coordinate Plane

Over Chapter 8

Find the geometric mean between 8 and 15. State the exact answer.

A.

B.

C.

D.

Over Chapter 8

A. yes

B. no

Determine whether the numbers 6, 9, and 12 are the measures of the sides of a right triangle.

Over Chapter 8

Find cos A for ΔABC if mC = 90, AB = x, AC = y, and CB = z.

A.

B.

C.

D.

Over Chapter 8

Find sin A for ΔABC if mC = 90, AB = x, AC = y, and CB = z.

A.

B.

C.

D.

Over Chapter 8

Find tan B for ΔABC if mC = 90, AB = x, AC = y, and CB = z.

A.

B.

C.

D.

Over Chapter 8

A. 26.5

B. 35.0

C. 51.3

D. 86.7

If mA = 42, AB = 7, and BC = 6 in ΔABC, what is mB?

You identified reflections and verified them as congruence transformations. (Lesson 4–7)

• Draw reflections.

• Draw reflections in the coordinate plane.

• line of reflection

Reflect a Figure in a Line

Draw the reflected image of quadrilateral WXYZ in line p.

Step 1 Draw segments perpendicular to line p from each point W, X, Y, and Z.

Step 2 Locate W', X', Y', and Z' so that line p is the perpendicular bisector of

Points W', X', Y', and Z' are the respective images of W, X, Y, and Z.

Reflect a Figure in a Line

Step 3 Connect vertices W', X', Y', and Z'.

Answer: Since points W', X', Y', and Z' are the images of points W, X, Y, and Z under reflection in line p, then quadrilateral W'X'Y'Z' is the reflection of quadrilateral WXYZ in line p.

Draw the reflected image of quadrilateral ABCD in line n.A. B.

C. D.

Minimize Distance by Using a Reflection

BILLIARDS Suppose that you must bounce the cue ball off side A before it rolls into the pocket at B. Locate the point C along side A that the ball must hit to ensure that it will roll directly toward the pocket.

Minimize Distance by Using a Reflection

Understand You are asked to locate a point C on side A such that the cue ball will bounce off to roll into the pocket at point B.

Plan For the cue ball to roll into the pocket at B, it must hit point C on side A somewhere in between where it sits now and the pocket at B. Use the reflection of point B on the continuation of the line that B lies on to help find this point.

Minimize Distance by Using a Reflection

Connect the cue ball with B' using a line. Locate point C at the intersection of the line drawn and side A.

Solve Draw point such that the corner pocket is the midpoint between B and B'.

Minimize Distance by Using a Reflection

Answer:

Check Check that BC B'C so that ΔBCB' is an isosceles triangle.

A. Determine how far the obstructing wall is from the ball.

B. Reflect point H over the line formed by wall W.

C. Determine the exact length of wall W.

D. Find the perpendicular distance from the hole to the wall.

MINIATURE GOLF Omar is playing miniature golf at a local course. Because a wall is blocking his direct shot, he needs to bounce the ball off wall W and hit the hole located at point H. Which of these steps would be needed to determine where on wall W Omar should aim?

Reflect a Figure in a Horizontal or Vertical Line

A. Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L(2, –1), and M(0, 1). Graph JKLM and its image over x = 1.

Reflect a Figure in a Horizontal or Vertical Line

Use the horizontal grid lines to find a corresponding point for each vertex so that each vertex and its image are equidistant from the line x = 1.

Answer:

Reflect a Figure in a Horizontal or Vertical Line

B. Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L(2, –1), and M(0, 1). Graph JKLM and its image over y = –2.

Reflect a Figure in a Horizontal or Vertical Line

Use the vertical grid lines to find a corresponding point for each vertex so that each vertex and its image are equidistant from the line y = –2.

Answer:

A. Quadrilateral ABCD has vertices A(1, 2), B(0, 1), C(1, –2), and D(3, 0). Graph ABCD and its image over x = 2.

A. B.

C. D.

B. Quadrilateral WXYZ has vertices W(2, 4), X(3, 3), Y(2, 0), and Z(0, 2). Graph WXYZ and its image over y = –1.

A. B.

C. D.

Reflect a Figure in the x- or y-axis

A. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its image reflected in the x-axis.

Multiply the y-coordinate of each vertex by –1.

(x, y) → (x, –y)

A(1, 1) → A'(1, –1)B(3, 2) → B'(3, –2)C(4, –1) → C'(4, 1)D(2, –3) → D'(2, 3)

Reflect a Figure in the x- or y-axis

Answer:

Reflect a Figure in the x- or y-axis

B. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its reflected image in the y-axis.

Multiply the x-coordinate of each vertex by –1.

(x, y) → (–x, y)

A(1, 1) → A'(–1, 1)B(3, 2) → B'(–3, 2)C(4, –1) → C'(–4, –1)D(2, –3) → D'(–2, –3)

Reflect a Figure in the x- or y-axis

Answer:

A. L'(3, –1), M'(5, –2), N'(6, 1), O'(4, 3)

B. L'(–3, 1), M'(–5, 2), N'(–6, –1), O'(–4, –3)

C. L'(–3, –1), M'(–5, –2), N'(–6, 1), O'(–4, 3)

D. L'(1, 3), M'(2, 5), N'(–1, 6), O'(–3, 4)

A. Graph quadrilateral LMNO with vertices L(3, 1), M(5, 2), N(6, –1), and O(4, –3) and its reflected image in the x-axis. Select the correct coordinates for the new quadrilateral L'M'N'O'.

B. Graph quadrilateral LMNO with vertices L(–1, 0), M(1, 1), N(2, –2), and O(0, –4) and its reflected image under the y-axis. Select the correct coordinates for the point M' in the new quadrilateral L'M'N'O'.

A. L'(–1, 0), M'(1, –1), N'(2, 2), O'(0, 4)

B. L'(1, 0), M'(–1, 1), N'(–2, –2), O'(0, –4)

C. L'(1, 0), M'(–1, –1), N'(–2, 2), O'(0, 4)

D. L'(0, –1), M'(1, 1), N'(–2, 2), O'(–4, 0)

Reflect a Figure in the Line y = x

Quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection of the line y = x.

Interchange the x- and y-coordinates of each vertex.

(x, y) → (y, x)

A(1, 1) → A'(1, 1)B(3, 2) → B'(2, 3)C(4, –1) → C'(–1, 4)D(2, –3) → D'(–3, 2)

Reflect a Figure in the Line y = x

Answer:

Quadrilateral EFGH has vertices E(–3, 1), F(–1, 3), G(1, 2), and H(–3, –1). Graph EFGH and its image under reflection of the line y = x. Select the correct coordinates for the point H' in the new quadrilateral E'F'G'H'.A. E'(–3, –1), F'(–1, –3),

G'(1, –2), H'(–3, 1)

B. E'(3, –1), F'(1, –3), G'(–1, 2), H'(3, –1)

C. E'(1, –3), F'(3, –1), G'(2, 1), H'(–1, –3)

D. E'(–1, 3), F'(–3, 1), G'(–2, –1), H'(1, 3)