Question

the rule ( d_{o, 0.25} (x, y) \to (0.25x, 0.25y) ) is applied to the segment lm to make an image of segment lm, not shown. the coordinates of l in the image are (square) (dropdown). the coordinates of m in the image are (square) (dropdown). the length, lm, is (square) (dropdown). the slope of the original segment and dilated segment are (square) (dropdown with options: both 0.25, both undefined, not the same, both zero).

Explanation:

Step 1: Find coordinates of L and M

From the graph, L is at \((-4, 8)\) and M is at \((4, 8)\).

Step 2: Find coordinates of L'

Apply the dilation rule \(D_{O, 0.25}(x, y) \to (0.25x, 0.25y)\) to L \((-4, 8)\):
\(x'\) = \(0.25\times(-4)\) = \(-1\)
\(y'\) = \(0.25\times8\) = \(2\)
So, L' is \((-1, 2)\).

Step 3: Find coordinates of M'

Apply the dilation rule to M \((4, 8)\):
\(x'\) = \(0.25\times4\) = \(1\)
\(y'\) = \(0.25\times8\) = \(2\)
So, M' is \((1, 2)\).

Step 4: Find length of L'M'

The distance between L' \((-1, 2)\) and M' \((1, 2)\) is calculated using the distance formula for horizontal lines (\(y\)-coordinates are the same):
Length = \(|x_2 - x_1|\) = \(|1 - (-1)|\) = \(|2|\) = \(2\). Alternatively, since the original length LM is \(|4 - (-4)| = 8\), and dilation factor is 0.25, length L'M' = \(8\times0.25 = 2\).

Step 5: Find slope of LM and L'M'

Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
For LM: \(m_{LM} = \frac{8 - 8}{4 - (-4)} = \frac{0}{8} = 0\)
For L'M': \(m_{L'M'} = \frac{2 - 2}{1 - (-1)} = \frac{0}{2} = 0\)
So, both slopes are zero.

Answer:

s:

• The coordinates of L' are \(\boldsymbol{(-1, 2)}\).
• The coordinates of M' are \(\boldsymbol{(1, 2)}\).
• The length of L'M' is \(\boldsymbol{2}\).
• The slope of the original segment and dilated segment are \(\boldsymbol{both\ zero}\).