# HOW TO GET THE SIDES OF A RIGHT TRIANGLE

## About "How to get the sides of a right triangle"

"How to get the sides of a right triangle" is the stuff related to forming a right triangle for which the lengths of the three sides are created.

For example, a teacher wants to give a question to the students related to right triangle.

She wants to create a question like "Find the missing side of the right triangle".

But she does not know, how to fix the lengths of the other two sides which can meet the condition of right triangle.

Here is the idea.

We can use the following formulas to form a right triangle.

Hypotenuse = m² + n²

Leg = m² - n²

Leg = 2mn

And always m > n

In the above formulas, you can take any positive values for "m" and "n" to get the three sides of a right triangle with the condition m > n.

For example, let m = 3, n = 2

Then,

hypotenuse   =   m² + n²   =   3² + 2²   =   9 + 4   =   13

leg   =   m² - n²   =   3² - 2²   =   9 - 4   =   5

leg   =   2mn   =   2(3)(2)   =   12

Now, we can check whether 13, 5, 12 be the three sides of the right angle triangle using Pythagorean theorem.

That is,

square of hypotenuse   =   sum of the squares of other sides

13²   =   5²   +   12²

169   =   25   +   144

169   =   169

In this way, we will be able to create the lengths of three sides of a right triangle using the formulas given above.

## Some more examples

Example 1 :

Using 5 and 6, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m = 6 and n = 5

Then, we have

Hypotenuse   =   m² + n²   =   6² + 5²   =   36 + 25   =   61

Leg   =   m² - n²   =   6² - 5²   =   36 - 25   =   11

Leg   =   2mn   =   2(6)(5)   =   60

Hence, the lengths of three sides of the right triangle are

61,  11  and  60

Let us look at the next example on "How to get the sides of a right triangle"

Example 2 :

Using 4 and 3, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m = 4 and n = 3

Then, we have

Hypotenuse   =   m² + n²   =   4² + 3²   =   16 + 9   =   25

Leg   =   m² - n²   =   4² - 3²   =   16 - 9   =   7

Leg   =   2mn   =   2(4)(3)   =   24

Hence, the lengths of three sides of the right triangle are

25,  7  and  24

Let us look at the next example on "How to get the sides of a right triangle"

Example 3 :

Using 1 and 2, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m = 2 and n = 1

Then, we have

Hypotenuse   =   m² + n²   =   2² + 1²   =   4 + 1   =   5

Leg   =   m² - n²   =   2² - 1²   =   4 - 1   =   3

Leg   =   2mn   =   2(2)(1)   =   4

Hence, the lengths of three sides of the right triangle are

5,  3  and  4

Let us look at the next example on "How to get the sides of a right triangle"

Example 4 :

Using 6 and 7, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m = 7 and n = 6

Then, we have

Hypotenuse   =   m² + n²   =   7² + 6²   =   49 + 36   =   85

Leg   =   m² - n²   =   7² - 6²   =   49 - 36   =   13

Leg   =   2mn   =   2(7)(6)   =   84

Hence, the lengths of three sides of the right triangle are

85,  13  and  84

Let us look at the next example on "How to get the sides of a right triangle"

Example 5 :

Using 11 and 6, create the lengths of three sides of a right triangle.

Solution :

Since m > n, we can take m = 11 and n = 6

Then, we have

Hypotenuse   =   m² + n²   =   11² + 6²   =   121 + 36   =   157

Leg   =   m² - n²   =   11² - 6²   =   121 - 36   =   85

Leg   =   2mn   =   2(11)(5)   =   110

Hence, the lengths of three sides of the right triangle are

157,  85  and  110

After having gone through the stuff, we hope that the students would have understood "how to get the sides of a right triangle".

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