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There are a few special right triangles that are worth knowing because they often pop up in problems like these. One very common one is the 3-4-5 right triangle. In this case, the difference between the legs isn’t 3cm (4-3 = 1). To make this triangle work, we must scale it up. Let’s double the sides, making it 6-8-10. Now, the difference between the legs is 2cm. To get to our desired difference of 3cm, let’s consider another example. We know that the sides must satisfy the Pythagorean theorem. To find the side lengths where the difference is 3cm, we need to test out different combinations. In our case, we know that two sides of our triangle are x and x + 3. The hypotenuse is h. Let’s go through a process of trial and error. Let’s pick some values and calculate to see if the equation holds. If x = 6, then the other side is 9. Using the Pythagorean theorem, 6² + 9² = 36 + 81 = 117. The square root of 117 is not a whole number. So, these sides won't work for this problem. When we are looking for whole number solutions, the trial-and-error method may take more time. Let’s assume the two sides are x and x + 3. Then h² = x² + (x + 3)². Since we are trying to find solutions where the difference between the sides is 3 cm, we can assume certain values. For example, if we have a right triangle with sides 5, 12, and 13, the difference between the sides isn’t 3cm. (12-5 = 7). Also, this set of numbers does not fit our conditions. The key is to try out the values, and see if the difference of 3 is satisfied. This step is a process of trying out values. While trial and error may be effective, it’s not the only way to solve the problem. Let’s look at another example: the sides are 12 and 5. With the hypotenuse of 13. The difference is 7. So, we're not quite there yet! However, by carefully selecting our values, we can find right triangles that satisfy the problem. The most important thing is to understand the relationship between the sides, which is given by the Pythagorean theorem.
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