Question Video: Using Proportions in a Triangle to Calculate an Unknown Length Mathematics • 11th Grade

Find the value of π‘₯.


Video Transcript

Find the value of π‘₯.

We see from the figure that the lines 𝐴𝐢 and 𝐴𝐡 are transversals that intersect parallel lines 𝐷𝐸 and 𝐡𝐢. And we know that the two pairs of corresponding angles created by this intersection are equal. That’s angles 𝐷𝐸𝐴 and 𝐡𝐢𝐴 and angles 𝐸𝐷𝐴 and 𝐢𝐡𝐴. This being the case, we can say the triangles 𝐴𝐡𝐢 and 𝐴𝐷𝐸 are similar triangles since they each have the common angle 𝐡𝐴𝐢 and their other two angles are also equal.

Now recall that when two triangles are similar, the ratios of the length of their corresponding sides are equal. In particular, 𝐴𝐷 is to 𝐴𝐡 as 𝐷𝐸 is to 𝐡𝐢. In other words, 𝐴𝐷 over 𝐴𝐡 is equal to 𝐷𝐸 over 𝐡𝐢. Now, we know that 𝐴𝐷 is equal to 10 units. 𝐴𝐡 is equal to 10 plus 11 units, that’s 𝐴𝐷 plus 𝐷𝐡. 𝐷𝐸 is equal to 10 units, and 𝐡𝐢 is π‘₯. And so we have 10 over 10 plus 11 is equal to 10 over π‘₯. That is, 10 over 21 is equal to 10 over π‘₯.

Now, solving for π‘₯, we multiply through by 21π‘₯ and divide both sides by 10, and we have π‘₯ equal to 21. Hence, using the given diagram, we find that π‘₯ is equal to 21 units.

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