Question

geometry units 1-6 study guide
unit 3: transformations
students will be able to transform figures using rigid t

1. triangle xyz is transformed to create triangle

xyz using the rule (x,y) → (x + b,y + c).

• triangle xyz has the vertices (3,2),(3,4),

and (0, -4) respectively.

• triangle xyz has the vertices

(-1,5),(-1,7), and (-4, -1) respectively.
what are the values of b and c?

b =

c =

Explanation:

Step1: Find \( b \) using x - coordinates

Take a vertex of \( \triangle XYZ \) (e.g., \( (3,2) \)) and its image \( (-1,5) \) in \( \triangle X'Y'Z' \). Using the transformation rule \( (x,y)\to(x + b,y + c) \), for the x - coordinate: \( 3 + b=-1 \). Solving for \( b \), we get \( b=-1 - 3=-4 \). We can verify with another vertex, say \( (3,4)\to(-1,7) \): \( 3 + b=-1\Rightarrow b = - 4 \), and \( (0,-4)\to(-4,-1) \): \( 0 + b=-4\Rightarrow b=-4 \).

Step2: Find \( c \) using y - coordinates

Using the same vertex \( (3,2)\to(-1,5) \), for the y - coordinate: \( 2 + c = 5 \). Solving for \( c \), we get \( c=5 - 2 = 3 \). Verify with \( (3,4)\to(-1,7) \): \( 4 + c=7\Rightarrow c = 3 \), and \( (0,-4)\to(-4,-1) \): \( - 4 + c=-1\Rightarrow c=3 \).

Answer:

\( b=-4 \)
\( c = 3 \)