[Triangle] Find the value of x and y.

[Triangle] Find the value of x and y.

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Nathanael Webber

When a line divides a triangle such that the line is parallel to a side of the triangle, then it divides the other sides proportionally.
The horizontal line is parallel to the base of the triangle so it divides the left and right sides of the triangle proportionally. Therefore:
$$\displaystyle\frac{{x}}{{16}}=\frac{{y}}{{20}}$$
The line with the double arrows is parallel to the right side of the triangle so it divides the left side and base proportionally. Therefore:
$$\displaystyle\frac{{x}}{{16}}=\frac{{45}}{{y}}$$
Since $$\displaystyle\frac{{x}}{{16}}=\frac{{y}}{{20}}$$ and $$\displaystyle\frac{{x}}{{16}}=\frac{{45}}{{y}}$$, then $$\displaystyle\frac{{y}}{{20}}=\frac{{45}}{{y}}$$. Solving for y gives: $$\displaystyle\frac{{y}}{{20}}=\frac{{45}}{{y}}$$
$$\displaystyle{y}^{{2}}={20}{\left({45}\right)}$$
$$\displaystyle{y}^{{2}}={900}$$
$$\displaystyle\sqrt{{{y}^{{2}}}}=\sqrt{{900}}$$
$$y=30$$
Substitute $$y=30$$ into either of the original proportions and solve for x:
$$\displaystyle\frac{{x}}{{16}}=\frac{{y}}{{20}}$$
$$\displaystyle\frac{{x}}{{16}}=\frac{{30}}{{20}}$$
$$\displaystyle\frac{{x}}{{16}}=\frac{{3}}{{2}}$$
$$\displaystyle{16}\cdot{\left(\frac{{x}}{{16}}\right)}={\left(\frac{{3}}{{2}}\right)}\cdot{16}$$
$$x=24$$