Problem 58862. Given Hypotenuse points create two right triangles

Given two points defining a hypotenuse create two right triangles of (h,5,R). Return the two (x,y) points that create the right triangles. I will elaborate on two geometric methods utilizing Matlab specific functions, rotation matrix, and translation matrix.
Given points [x1,y1] and [x2,y2] return [x3 y3;x4 y4] such that distance(xy2,xy3)=distance(xy2,xy4)=5. h>5
The below figure is created based upon h=distance([x1,y1],[x2,y2]), translating (x1,y1) to (0,0), and rotating (x2,y2) to be on the Y-axis. From this manipulation two right triangles are apparent: [X,Y,R] and [X,h-Y,5] with R^2+5^2=h^2. Subtracting and simplifying these triangles leads to Y and two X values after substituting back into R^2=X^+Y^2 equation.
P^2=X^2+(h-Y)^2 and R^2=X^2+Y^2 after subtraction gives R^2-P^2=Y^2-(d-Y)^2 = Y^2-d^2+2dY-Y^2=2dY-d^2 thus
Y=(R^2-P^2+h^2)/(2h) and X=+/- (R^2-Y^2)^.5
The trick is to now un-rotate and translate this solution matrix using t=atan2(dx,dy), [cos(t) -sin(t);sin(t) cos(t)] and [x1 y1]
A second method to find (X,Y) is theta=atan(5/R), X=Rsin(theta) and Y=Rcos(theta). The rotation and translation matrices are still required to return to the original coordinate system.
In this figure h represents distance from (x1,y1) to (x2,y2) and (x1,y1) has been translated to 0,0
Solve

Solution Stats

100.0% Correct | 0.0% Incorrect
Last Solution submitted on Nov 18, 2024

Solution Comments

Show comments

Problem Recent Solvers5

Suggested Problems

More from this Author308

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!