To visualize and create the triangle LMN with the specified properties, follow the instructions below:
A. Creating Triangle LMN
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Choose Coordinates for L, M, and N:
- Let \( L(0, 3) \) (this point will be the top of the vertical line)
- Let \( M(4, 3) \) (this point will be to the right of L on the same horizontal line)
- Let \( N(0, 0) \) (this point will be at the bottom of the vertical line)
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Plot the Triangle:
- Draw the vertical line segment \( LN \) from \( L \) to \( N \).
- Draw the horizontal line segment \( MN \) from \( M \) to \( N \).
- Draw the line segment \( LM \) from \( L \) to \( M \).
B. Creating QR Parallel to LM
- Choose Coordinates for Points Q and R:
- Let \( Q(0, 1) \) (this point will be located somewhere on the vertical line segment \( LN \))
- Let \( R(4, 1) \) (this point will be directly to the right of Q on the horizontal line)
C. Labeling Points and Their Coordinates
- \( L(0, 3) \)
- \( M(4, 3) \)
- \( N(0, 0) \)
- \( Q(0, 1) \)
- \( R(4, 1) \)
Slope Calculation
To confirm that line segment \( QR \) is parallel to line segment \( LM \), we need to calculate the slopes of both line segments using the slope formula, which is given by:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
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Calculate the Slope of LM:
- Using points \( L(0, 3) \) and \( M(4, 3) \): \[ \text{slope of LM} = \frac{3 - 3}{4 - 0} = \frac{0}{4} = 0 \]
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Calculate the Slope of QR:
- Using points \( Q(0, 1) \) and \( R(4, 1) \): \[ \text{slope of QR} = \frac{1 - 1}{4 - 0} = \frac{0}{4} = 0 \]
Conclusion
Since the slopes of both line segments \( QR \) and \( LM \) are equal (both are 0), we can confirm that line \( QR \) is indeed parallel to line \( LM \).
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