Step by step
The question
Question: find the distance between (−2, 5) and (3, −1).
1 Plot the points
Draw the grid.
Put both points on the grid: (−2, 5)
and (3, −1).
The distance is the length of this line.
We call it d.
2 Make a right triangle
From the left point, go across.
From x = −2 to x = 3 is 5.
We call it a = 5.
Then go down.
From y = 5 to y = −1 is 6.
A length is never negative.
So b = 6.
Across and down make a right angle.
The distance d is the longest side.
3 Use a² + b² = c²
The rule for every right triangle: a² + b² = c².
Here c is the distance d.
Now put the numbers in: a = 5 and b = 6.
5² means 5 times 5.
That is 25.
6² means 6 times 6.
That is 36.
Write the squares: 25 + 36 = d².
25 plus 36 is 61.
d² is 61.
So d is the square root of 61.
61 is not a square number.
So we keep √61.
That is the exact answer.
With a calculator, √61 is about 7.8, to the nearest tenth.
4 Check
Let's check with the distance formula.
It does the same steps in one line.
Put in the numbers of (3, −1) and (−2, 5).
3 minus −2 is 5.
Taking away a negative is adding.
−1 minus 5 is −6.
Careful: (−6)² is −6 times −6.
Minus times minus is plus: 36, not −36.
25 plus 36 is 61.
We get √61 again.
Right!
Let's look back.
First: plot the points.
Then: make a right triangle.
Then: use a² + b² = c².
Last: check.