Question Video: Using the Properties of Parallel Lines to Solve a Problem Mathematics • 11th Grade

In the figure, lines 𝐿₁, 𝐿₂, 𝐿₃, and 𝐿₄ are all parallel. Given that 𝑋𝑍 = 12, 𝑍𝑁 = 8, 𝐴𝐡 = 10, and 𝐡𝐢 = 5, what is the length of line segment 𝐢𝐷?


Video Transcript

In the figure, lines 𝐿 sub one, 𝐿 sub two, 𝐿 sub three, and 𝐿 sub four are parallel. Given that 𝑋𝑍 equals 12, 𝑍𝑁 equals eight, 𝐴𝐡 equals 10, and 𝐡𝐢 equals five, what is the length of line segment 𝐢𝐷?

Let’s start by filling in the measurements of the line segments that we’re given. We can see in the diagram that there are four parallel lines, 𝐿 one to 𝐿 four, and there are two transversals, 𝑀 and 𝑀 prime. A transversal is a line which passes through two lines in the same plane at two distinct points.

To find the missing length of 𝐢𝐷, we’re going to use a key fact about parallel lines and the proportionality of the transversals. And that is, if three or more parallel lines are cut by two transversals. Then, they divide the transversal proportionally. So taking a look on our transversal 𝑀 prime, we can write that our segment 𝑍𝑁 over the segment 𝑋𝑍 has a proportional relationship to the transversal 𝑀, which is equal to the segment 𝐢𝐷 over the segment 𝐴𝐢.

If we define our unknown length 𝐢𝐷 as π‘₯, then we can fill in the values that we have for our line segments. Since 𝑍𝑁 is eight and 𝑋𝑍 is 12, we can write eight over 12. On the transversal 𝑀, we have our unknown length 𝐢𝐷 as π‘₯. And the line 𝐴𝐢 is made up of the sections five and 10, which is 15. We can simplify our fraction eight 12ths by dividing the numerator and denominator by four, which leaves us with the equation two-thirds equals π‘₯ over 15.

To solve this, we take the cross product. So, we have two times 15 which is 30 is equal to three π‘₯. And therefore, to find π‘₯, we divide both sides by three, which gives us that 10 equals π‘₯, where π‘₯ is 10. And since π‘₯ corresponds to the length of 𝐢𝐷, then we have our final answer that the length of 𝐢𝐷 is 10.

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