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# Find the lenth

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In the diagram below, we have $ST = 2TR$ and $PQ = SR = 20$. Find the length $UV$.

Mar 30, 2020

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Hmmm, I couldn't find a geometric way to solve this within the 1 minute I looked at this problem. I am very inpatient!

So let us use coordinates!

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Find ST

ST = 40/3 by simple algebra I think you can do at your skill level judging by the difficult of this problem.

Let QR = $$a$$

Slope of PT: $$-\frac{\frac{40}{3}}{a}$$

Y-intercept of PT: 20

Equation: $$y=-\frac{\frac{40}{3}}{a}x+20$$

Slope of QS: (20/a)

Y intercept of QS: 0

Equation: $$y=\frac{20}{a}x$$

Substitute

$$-\frac{\frac{40}{3}}{a}x+20=\frac{20}{a}x$$

$$20 =\frac{\frac{100}{3}}{a}x$$

$$20a=\frac{100x}{3}$$

$$60a=100x$$

$$3a=5x$$

$$\frac{5}{3}x=a$$

Interpret this:

$$x$$ is the x-coordinate of the solution of we solved for the location of the intercept at point U. That means $$x$$ is the length of QV. We know $$a$$ is the length of QR.

That means QV is three-fifths the length of QR.

We know that QUV is similar to QSR by AA similarity through a series of proofs.

Since we know the proportion of the sides, we can solve for UV:

Solve:

20 * (3/5) = 12

Ta-da! Mathz!

Mar 30, 2020