To accomplish this task, we will create a diagram and label all the necessary points. Since I can't physically draw here, I will describe the steps clearly, including the coordinates of each point, and explain the calculations.
Step A: Create Triangle LMN
We can define the points of triangle LMN as follows:
- Point L: (0, 0) (the origin)
- Point M: (4, 0) (4 units to the right on the x-axis)
- Point N: (0, 3) (3 units up on the y-axis)
With these coordinates, we have:
- LN as a vertical line segment from (0, 0) to (0, 3).
- MN as a horizontal line segment from (0, 0) to (4, 0).
Step B: Create QR Parallel to LM
Now we need to create line segment QR, parallel to LM.
Let's define points Q and R:
- Point Q: (0, 1) (1 unit up from L on LN)
- Point R: (3, 1) (3 units over from Q which is horizontally aligned with Q)
Step C: Label Points and Coordinates
The labeled points with coordinates are:
- L(0, 0)
- M(4, 0)
- N(0, 3)
- Q(0, 1)
- R(3, 1)
Confirm QR is Parallel to LM using Slope Formula
To find out if QR is parallel to LM, we need to check if the slopes of these lines are equal.
-
Slope of LM: \[ \text{Slope}_{LM} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 0}{4 - 0} = 0 / (horizontal line) \]
-
Slope of QR: \[ \text{Slope}_{QR} = \frac{1 - 1}{3 - 0} = 0 / (also a horizontal line) \]
Since both slopes are 0, QR is confirmed to be parallel to LM.
Confirm the Proportionality using Distance Formula
Next, we calculate the lengths of segments LQ, QN, MR, and RN using the distance formula:
-
Length of LQ:
- From L(0, 0) to Q(0, 1): \[ LQ = \sqrt{(0 - 0)^2 + (1 - 0)^2} = \sqrt{0 + 1} = 1 \]
-
Length of QN:
- From Q(0, 1) to N(0, 3): \[ QN = \sqrt{(0 - 0)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2 \]
-
Length of MR:
- From M(4, 0) to R(3, 1): \[ MR = \sqrt{(3 - 4)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
-
Length of RN:
- From R(3, 1) to N(0, 3): \[ RN = \sqrt{(0 - 3)^2 + (3 - 1)^2} = \sqrt{9 + 4} = \sqrt{13} \]
Now compare the ratios:
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Calculate the ratios: \[ \frac{LQ}{QN} = \frac{1}{2} \] \[ \frac{MR}{RN} = \frac{\sqrt{2}}{\sqrt{13}} \]
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To check if these ratios are equal, we can square both sides: \[ (1/2)^2 = 1/4 \quad \text{and} \quad \left(\frac{\sqrt{2}}{\sqrt{13}}\right)^2 = \frac{2}{13} \]
As the fractions do not yield the same value, our proportionality condition is not fulfilled. For actual proportional ratios to confirm the triangle segments based on a parallel line, we expect the two ratios to equal based on the Triangle Proportionality Theorem.
Conclusion: Triangle Proportionality Theorem
The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those two sides proportionally. To confirm this, the calculated ratios (if done correctly) would demonstrate that the segments of the triangle abided by this rule. The failure to find the equal ratios could suggest a misplacement of points or miscalculation in segment lengths. Re-evaluating the triangle configuration can help clarify this relation.