Worked examples
The question
Question: find the hypotenuse of a 45-45-90 triangle with a leg of 5.
1 Half of a square
The angles are 45°, 45° and 90°.
This is half of a square.
The longest side cuts the square in two.
So the two legs are sides of the square.
They are equal: both are 5.
2 Use the pattern
The pattern: the hypotenuse is a leg times √2.
Here the leg is 5.
So c = 5 × √2.
c = 5 × √2
Why √2?
Pythagoras: x² + x² is 2x².
The square root of 2x² is x times √2.
x² + x² = 2x² → c = x√2
We leave out the times sign.
So c = 5√2.
c = 5√2
That is about 7.1.
≈ 7.1
Careful!
The hypotenuse is not the two legs added.
The way along the diagonal is shorter than going round two sides.
Careful: not 5 + 5 = 10 ✗
3 Check
Let's check with Pythagoras: a² + b².
52 + 52 = 25 + 25 = 50
Now c²: (5√2)² is 5² times 2.
(5√2)2 = 25 × 2 = 50
Both are 50.
So c = 5√2 is right!
Let's look back.
First: half of a square.
Then: use the pattern.
Last: check.
▶ Watch it solved
The question
Question: find the legs of a 45-45-90 triangle with a hypotenuse of 8.
1 Half of a square
The angles are 45°, 45° and 90°.
This is half of a square.
The longest side cuts the square in two.
So the two legs are sides of the square.
They are equal.
We call both x.
2 Use a² + b² = c²
Both legs are x.
So Pythagoras gives x² + x² = 8².
x2 + x2 = 82
8² means 8 times 8.
That is 64.
8² = 8 × 8 = 64
One x² and one more x² make 2x².
x² + x² = 2x²
So 2x² = 64.
2x² = 64
Careful!
The legs are not half of the hypotenuse.
The hypotenuse is a leg times √2, not times 2.
Careful: not x = 8 ÷ 2 ✗
3 Remove the 2
2x² means 2 times x².
To remove times 2, we divide by 2.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
2x22 = 642
2 divided by 2 is 1.
So just x² is left.
64 divided by 2 is 32.
x2 = 32
4 Take the square root
x times x is 32.
So x is the square root of 32.
x = √32
But 32 has a square number in it: 16.
32 is 16 times 2.
x = √16 × 2
A root of a times splits into two roots.
x = √16 × √2
√16 is 4.
4 × 4 = 16
So x = 4√2.
That is the exact answer, made simple.
Both legs are 4√2.
x = 4√2
5 Check
Let's check: put x = 4√2 in x² + x².
(4√2)2 + (4√2)2 = 32 + 32 = 64
We get 64, the same as 8².
So x = 4√2 is right!
Let's look back.
First: half of a square.
Then: use a² + b² = c².
Then: remove the 2.
Then: take the square root.
Last: check.
▶ Watch it solved
The question
Question: find the legs of a 45-45-90 triangle with a hypotenuse of 12.
1 Half of a square
The angles are 45°, 45° and 90°.
This is half of a square.
The longest side cuts the square in two.
So the two legs are sides of the square.
They are equal.
We call both x.
2 Use a² + b² = c²
Both legs are x.
So Pythagoras gives x² + x² = 12².
x2 + x2 = 122
12² means 12 times 12.
That is 144.
12² = 12 × 12 = 144
One x² and one more x² make 2x².
x² + x² = 2x²
So 2x² = 144.
2x² = 144
Careful!
The legs are not half of the hypotenuse.
The hypotenuse is a leg times √2, not times 2.
Careful: not x = 12 ÷ 2 ✗
3 Remove the 2
2x² means 2 times x².
To remove times 2, we divide by 2.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
2x22 = 1442
2 divided by 2 is 1.
So just x² is left.
144 divided by 2 is 72.
x2 = 72
4 Take the square root
x times x is 72.
So x is the square root of 72.
x = √72
But 72 has a square number in it: 36.
72 is 36 times 2.
x = √36 × 2
A root of a times splits into two roots.
x = √36 × √2
√36 is 6.
6 × 6 = 36
So x = 6√2.
That is the exact answer, made simple.
Both legs are 6√2.
x = 6√2
5 Check
Let's check: put x = 6√2 in x² + x².
(6√2)2 + (6√2)2 = 72 + 72 = 144
We get 144, the same as 12².
So x = 6√2 is right!
Let's look back.
First: half of a square.
Then: use a² + b² = c².
Then: remove the 2.
Then: take the square root.
Last: check.
▶ Watch it solved
