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The Diagram

The blue dot moves, I know its x and y coordinates, and I also know that that causes the red dot to move along the triangle's hypotenuse in such a way that the red dot's y coordinate is always equal to the blue dot's y-coordinate.

How do I calculate the x value of the red dot's coordinate?

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    http://en.wikipedia.org/wiki/Dot_product#Geometric_interpretation2011-12-12

2 Answers 2

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Making the assumptions stated in the comments, and further that the lower leg of the triangle is horizontal as pictured, and $m$ is the slope of the hypotenuse.

Then $y_{\text{red}}/x_{\text{red}} = m$, $y_{\text{red}} = y_{\text{blue}}$. Solving gives $x_\text{red} = y_\text{blue}/m$.

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Assume the base of the triangle is horizontal at with the vertex at $(0,0)$ and length $x$. Furthermore, the triangle is right with a height $h$. Define the coordinates of the blue point as $(x_1,y)$ and the red point as $(x_2,y)$. Then, we want $x_2$ as a function of the common y-cordinate.

From basic trig, the interior angle between side x and the hypotenuse is given by $sin^{-1}\left(\frac{h}{x}\right)=\theta$. Note that when the the blue dot is at any height $y$ , $0\leq y\leq h$, drawing a vertical line from the red dot to the base forms a similar triangle with height y and base $x_2$. The angle between the base and hypotenuse will simply be $\theta$. Thus, we have from basic trig:$sin(\theta) = \frac{y}{x_2} \Rightarrow sin\left(sin^{-1}\left(\frac{h}{x}\right)\right) = \frac{y}{x_2} \Rightarrow x_2 = \frac{y\cdot h}{x}\;\;\;0\leq y \leq h$ Which gives the x-cordinate of the red point in terms of the givens.