$Why if $\frac{opp}{adj} =x$, then $x \times hyp =$ The length of a line perpendicular to the hypotenuse with the same height.$

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$Why if $\frac{opp}{adj} =x$, then $x \times hyp =$ The length of a line perpendicular to the hypotenuse with the same height.$

Why if $\frac{opp}{adj} =x$, then

$x \times hyp =$ The length of a line perpendicular to the hypotenuse with the same height.

So if you have a triangle and you divide the opposite by the adjacent, the ratio you get can be used to multiply the hypotenuse to get the length of a line perpendicular to the hypotenuse with the same height.

What exactly is happening why does this work?

See the file attached for a more visual representation.

Geometry

Math Gnome

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$Matchmaticians Why if $\frac{opp}{adj} =x$, then  $x \times hyp =$ The length of a line perpendicular to the hypotenuse with the same height. File #1$ File #1 (png)

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Let $D$ be the point where the green line in your picture intersects the $x$-axis. Then the two traingles $ABC$ and $ADC$ are similar. So we have
\[x=\frac{AB}{BC}=\frac{AD}{AC},\]
note that $\frac{AB}{BC}$ and $\frac{AD}{AC}$ are $\frac{opp}{adj}$ for theangle $C$ in traingles $ABC$ and $ADC$, respectively.

Hence
\[x=\frac{AD}{AC} \Rightarrow AD=x\cdot AC,\]
which is what you wanted to prove.

Erdos

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$Why if $\frac{opp}{adj} =x$, then $x \times hyp =$ The length of a line perpendicular to the hypotenuse with the same height.$

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