How would you determine the lengths of all sides of a similar right triangle that is 1" offset from all sides?

Question

Given a right triangle with a base (side A) of 44", a height (side B) of 22" and a hypotenuse (side C) of 49.1935" (based on Pythagorean theorem).

How would I determine the lengths of each side of a smaller similar triangle that is exactly 1" smaller offset on all sides?

In this case, the base (side A) would be 38.7639", the height (side B) would be 19.382" and the hypotenuse (side C) would be 43.3394".

Answer 1

Hi Andrew,

 

Here are some hints (assuming you have not already figured it out). Use some coordinate geometry, assigning coordinates to the vertices of the triangle, write some equations for the sides of the outside triangle and this "inside" triangle. The difficult part is finding the equation of the hypotenuse of the inside triangle. 

 

Points for the outside triangle are (0,0), (0,22), (44,0). The coordinates of the right angle for the inside triangle will be (1,1). Finding the other two vertices of the inside triangle is the key. Then you can use the distance formula. Here is a graph and some equations for you to look over. 

 

https://www.desmos.com/calculator/xmsnnghojx

 

There are probably different ways to do this. Consider any point on the hypotenuse of the outside triangle, e.9. (6,19). After finding the equation of the outside hypotenuse you can write the equation for a perpendicular to that line passing through your random point on the hypotenuse. If you then think of a circle with radius one centered at (6,19), one of the points where it intersects the perpendicular will be a point that a line parallel to the outside hypotenuse also contains. Therefore you can write the equation of the inside hypotenuse. From there you can find the other two vertices of the inside triangle, then use the distance formula to find the lengths of the 3 sides (or use similar triangle ratios). 

 

Good luck writing that in JavaScript.