Find X-coordinate of vertex, when Y-coordinate is given along with the other two vertices in a triangle

Question

I need to find the X-coordinate of a vertex in a right triangle. It's been a while since I used trigonometry and I can't seem to wrap my head around it.

Image with coordinates

The Y-coordinate is always the same as the Y-coordinate of the vertex on the left, under the radius. I've tried using line intersection, but I don't know how to use that here since I don't know where to put the X coordinate of the diagonal line. What method can I use to solve this?

Answer 1

If you have two points in a right triangle and the y-coordinate of the third point, you can use the Pythagorean theorem and the slope of the triangle's hypotenuse to find the x-coordinate of the third point. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's say you have two points A(x1, y1) and B(x2, y2) in the triangle, and the y-coordinate of the third point is y3. To find the x-coordinate of the third point, follow these steps:

1. Calculate the slope of the line AB: Slope (m) = (y2 - y1) / (x2 - x1)

2. Since the triangle is a right triangle, the slope of the line perpendicular to AB (the hypotenuse) is the negative reciprocal of the slope of AB: Perpendicular slope = -1 / m

3. Use the perpendicular slope and the given y-coordinate (y3) to find the equation of the line perpendicular to AB: Equation: y - y3 = perpendicular slope * (x - x3)

4. Use the Pythagorean theorem to set up an equation involving the distances between the points: (x2 - x1)^2 + (y2 - y1)^2 = (x3 - x1)^2 + (y3 - y1)^2

5. Substitute the equation from step 3 into the equation from step 4 and solve for x3.

This process should give you the x-coordinate of the third point in the right triangle.

The formula for calculating the x-coordinate (x3) of the third point in a right triangle given two points (x1, y1) and (x2, y2), and the y-coordinate (y3) of the third point is:

Image of formula

This formula uses the slope of the line connecting the two known points to calculate the x-coordinate of the third point where it intersects the line with the given y-coordinate.