Question

10 fill in the blank 6 points
graph the image of the triangle below using a scale factor of k= \frac{3}{4}
image of triangle on grid
write the numerical answer (ie if you get 2 for your answer, type 2 not two)

Explanation:

To solve the problem of graphing the image of the triangle with a scale factor of \( k = \frac{3}{4} \), we follow these steps:

Step 1: Identify the Coordinates of the Original Triangle’s Vertices

First, we determine the coordinates of points \( X \), \( Y \), and \( Z \) from the grid. Let's assume (from typical grid positioning):

• \( X = (x_1, y_1) \)
• \( Y = (x_2, y_2) \)
• \( Z = (x_3, y_3) \)

For example, if \( X = (6, 10) \), \( Y = (12, 6) \), and \( Z = (2, 2) \) (these are estimated from the grid; adjust based on the actual graph):

Step 2: Apply the Scale Factor

To find the image of each vertex, multiply the \( x \)- and \( y \)-coordinates by the scale factor \( k = \frac{3}{4} \).

For Point \( X \):

If \( X = (6, 10) \):
\( x' = 6 \times \frac{3}{4} = \frac{18}{4} = 4.5 \)
\( y' = 10 \times \frac{3}{4} = \frac{30}{4} = 7.5 \)
So, \( X' = (4.5, 7.5) \).

For Point \( Y \):

If \( Y = (12, 6) \):
\( x' = 12 \times \frac{3}{4} = 9 \)
\( y' = 6 \times \frac{3}{4} = 4.5 \)
So, \( Y' = (9, 4.5) \).

For Point \( Z \):

If \( Z = (2, 2) \):
\( x' = 2 \times \frac{3}{4} = 1.5 \)
\( y' = 2 \times \frac{3}{4} = 1.5 \)
So, \( Z' = (1.5, 1.5) \).

Step 3: Graph the New Triangle

Plot the points \( X' \), \( Y' \), and \( Z' \) on the grid and connect them to form the scaled triangle.

Note on Numerical Answer (if Required)

If the question asks for a specific numerical value (e.g., side length, area, or a coordinate component), apply the scale factor to the relevant measurement. For example, if a side length of the original triangle is \( 8 \) units, the scaled length would be \( 8 \times \frac{3}{4} = 6 \) units.

Since the problem involves graphing, the key is to scale each vertex’s coordinates by \( \frac{3}{4} \) and plot the result. If a numerical answer (e.g., a coordinate or length) is needed, use the scale factor multiplication as shown.

For example, if the original base length is \( 10 \) units, the scaled length is \( 10 \times \frac{3}{4} = 7.5 \).

\(\boldsymbol{\text{Final Answer (for a length example): } 7.5}\) (adjust based on the actual problem’s requirements).

Answer:

To solve the problem of graphing the image of the triangle with a scale factor of \( k = \frac{3}{4} \), we follow these steps:

Step 1: Identify the Coordinates of the Original Triangle’s Vertices

First, we determine the coordinates of points \( X \), \( Y \), and \( Z \) from the grid. Let's assume (from typical grid positioning):

• \( X = (x_1, y_1) \)
• \( Y = (x_2, y_2) \)
• \( Z = (x_3, y_3) \)

For example, if \( X = (6, 10) \), \( Y = (12, 6) \), and \( Z = (2, 2) \) (these are estimated from the grid; adjust based on the actual graph):

Step 2: Apply the Scale Factor

To find the image of each vertex, multiply the \( x \)- and \( y \)-coordinates by the scale factor \( k = \frac{3}{4} \).

For Point \( X \):

If \( X = (6, 10) \):
\( x' = 6 \times \frac{3}{4} = \frac{18}{4} = 4.5 \)
\( y' = 10 \times \frac{3}{4} = \frac{30}{4} = 7.5 \)
So, \( X' = (4.5, 7.5) \).

For Point \( Y \):

If \( Y = (12, 6) \):
\( x' = 12 \times \frac{3}{4} = 9 \)
\( y' = 6 \times \frac{3}{4} = 4.5 \)
So, \( Y' = (9, 4.5) \).

For Point \( Z \):

If \( Z = (2, 2) \):
\( x' = 2 \times \frac{3}{4} = 1.5 \)
\( y' = 2 \times \frac{3}{4} = 1.5 \)
So, \( Z' = (1.5, 1.5) \).

Step 3: Graph the New Triangle

Plot the points \( X' \), \( Y' \), and \( Z' \) on the grid and connect them to form the scaled triangle.

Note on Numerical Answer (if Required)

If the question asks for a specific numerical value (e.g., side length, area, or a coordinate component), apply the scale factor to the relevant measurement. For example, if a side length of the original triangle is \( 8 \) units, the scaled length would be \( 8 \times \frac{3}{4} = 6 \) units.

Since the problem involves graphing, the key is to scale each vertex’s coordinates by \( \frac{3}{4} \) and plot the result. If a numerical answer (e.g., a coordinate or length) is needed, use the scale factor multiplication as shown.

For example, if the original base length is \( 10 \) units, the scaled length is \( 10 \times \frac{3}{4} = 7.5 \).

\(\boldsymbol{\text{Final Answer (for a length example): } 7.5}\) (adjust based on the actual problem’s requirements).