Worked examples
The question
Question: find out if a triangle with sides 3, 4, 5 is a right triangle.
1 Find the longest side
In a right triangle, the longest side is across from the right angle.
We call it c.
Here it is 5.
The two shorter sides are a and b.
Every right triangle has a² + b² = c².
So we test it.
So c = 5, the longest side.
c = 5
2 Square the short sides
Now put the numbers in: a = 3 and b = 4.
a2 + b2 = 32 + 42
3² means 3 times 3.
That is 9.
32 = 3 × 3 = 9
4² means 4 times 4.
That is 16.
42 = 4 × 4 = 16
Write the squares.
a2 + b2 = 9 + 16
9 plus 16 is 25.
So a² + b² is 25.
a2 + b2 = 25
3 Square the longest side
Now put the numbers in: c = 5.
c2 = 52
5² means 5 times 5.
That is 25.
52 = 5 × 5 = 25
So c² is 25.
c2 = 25
Careful!
c is always the longest side.
Find it first.
Careful: c is always the longest side
4 Compare
a² + b² is 25.
c² is 25.
They are the same!
25 = 25
The rule a² + b² = c² works only in right triangles.
So the angle across from c is a right angle.
So yes: it is a right triangle.
= Yes: a right triangle
Let's look back.
First: find the longest side.
Then: square the short sides.
Then: square the longest side.
Last: compare.
Answer = Yes: a right triangle
▶ Watch it solved
The question
Question: find out if a triangle with sides 13, 5, 12 is a right triangle.
1 Find the longest side
In a right triangle, the longest side is across from the right angle.
We call it c.
Here it is 13.
The two shorter sides are a and b.
Every right triangle has a² + b² = c².
So we test it.
So c = 13, the longest side.
c = 13
2 Square the short sides
Now put the numbers in: a = 5 and b = 12.
a2 + b2 = 52 + 122
5² means 5 times 5.
That is 25.
52 = 5 × 5 = 25
12² means 12 times 12.
That is 144.
122 = 12 × 12 = 144
Write the squares.
a2 + b2 = 25 + 144
25 plus 144 is 169.
So a² + b² is 169.
a2 + b2 = 169
3 Square the longest side
Now put the numbers in: c = 13.
c2 = 132
13² means 13 times 13.
That is 169.
132 = 13 × 13 = 169
So c² is 169.
c2 = 169
Careful!
c is the longest side, 13.
Not the last number we read.
Careful: not 132 + 52 = 122 ✗
4 Compare
a² + b² is 169.
c² is 169.
They are the same!
169 = 169
The rule a² + b² = c² works only in right triangles.
So the angle across from c is a right angle.
So yes: it is a right triangle.
= Yes: a right triangle
Let's look back.
First: find the longest side.
Then: square the short sides.
Then: square the longest side.
Last: compare.
Answer = Yes: a right triangle
▶ Watch it solved
The question
Question: find out if a triangle with sides 25, 7, 24 is a right triangle.
1 Find the longest side
In a right triangle, the longest side is across from the right angle.
We call it c.
Here it is 25.
The two shorter sides are a and b.
Every right triangle has a² + b² = c².
So we test it.
So c = 25, the longest side.
c = 25
2 Square the short sides
Now put the numbers in: a = 7 and b = 24.
a2 + b2 = 72 + 242
7² means 7 times 7.
That is 49.
72 = 7 × 7 = 49
24² means 24 times 24.
That is 576.
242 = 24 × 24 = 576
Write the squares.
a2 + b2 = 49 + 576
49 plus 576 is 625.
So a² + b² is 625.
a2 + b2 = 625
3 Square the longest side
Now put the numbers in: c = 25.
c2 = 252
25² means 25 times 25.
That is 625.
252 = 25 × 25 = 625
So c² is 625.
c2 = 625
Careful!
c is the longest side, 25.
Not the last number we read.
Careful: not 252 + 72 = 242 ✗
4 Compare
a² + b² is 625.
c² is 625.
They are the same!
625 = 625
The rule a² + b² = c² works only in right triangles.
So the angle across from c is a right angle.
So yes: it is a right triangle.
= Yes: a right triangle
Let's look back.
First: find the longest side.
Then: square the short sides.
Then: square the longest side.
Last: compare.
Answer = Yes: a right triangle
▶ Watch it solved
