Find the dimensions of all four right triangles

Question

A rectangle encloses a right triangle as shown. The length of one side of the inner triangle is 25 units. There are four right triangles in the figure, all of which have sides the lengths of which are all integers. Find the dimensions of all the triangles. The diagram is not drawn to scale. There may be more than one solution!

Answer 1

Sir,

If we agree to this,

"There are four right triangles in the figure, all of which have sides the lengths of which are all integers"

then this question has no solution.

Edit:

I found the solution

https://ibb.co/tM9thBZ

please check

 

Comments

It has at least one solution. There’s a clue: 25 is the hypotenuse of one triangle and the other two sides have to be integers. Think about how to construct a Pythagorean triangle with integer sides. For example, 3-4-5 is probably the most well known one. Work on the difference between the squares of two integers and you’ll see how to make many Pythagorean triangles with integer sides. Also bear in mind the triangles are inside a rectangle which also has integer sides, and opposite sides of the rectangle have equal length.

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If you really would like to know the answer, I can send you a private message and show you the step-by-step logic of how you get the answer.

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Can you tell me by looking at the picture, am I completely off the track?

https://ibb.co/7j74G0y

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You are off track, but I can see you have the right idea. You have made an assumption which limits your choice of the sides of ABC. Here’s a clue. You are right to start with ABC, but look at it like this:

(AC-BC)(AC+BC)=25², then consider the factors of 25². If the factors are p and q:

AC-BC=p (you assumed p=1 and q=625 are the only factors).

AC+BC=q, so AC=(p+q)/2, BC=(q-p)/2. This will give you the dimensions of ABC. Note also that there are two similar triangles which will help you resolve ambiguities. There are in fact two solutions.

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I am not giving up. But this question is hard for me. I am brainstorming but without any results :(

I was able to find PAB and QBC is similar and BC= 60 and AC =65.

This is all for now, but I am still on it.

Thank You Sir.

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Well done! Your hard work was rewarded. You didn’t give up, and that’s good!