Step by step
The question
Question: find the distance between (−7, 3) and (5, −2).
1 Plot the points
Draw the grid.
Put both points on the grid: (−7, 3)
and (5, −2).
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 = −7 to x = 5 is 12.
We call it a = 12.
Then go down.
From y = 3 to y = −2 is 5.
A length is never negative.
So b = 5.
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 = 12 and b = 5.
12² means 12 times 12.
That is 144.
5² means 5 times 5.
That is 25.
Write the squares: 144 + 25 = d².
144 plus 25 is 169.
d² is 169.
So d is the square root of 169.
13 times 13 is 169.
So d = 13.
The distance is 13.
4 Check
Let's check with the distance formula.
It does the same steps in one line.
Put in the numbers of (5, −2) and (−7, 3).
5 minus −7 is 12.
Taking away a negative is adding.
−2 minus 3 is −5.
Careful: (−5)² is −5 times −5.
Minus times minus is plus: 25, not −25.
144 plus 25 is 169.
The square root is 13 again.
Right!
Let's look back.
First: plot the points.
Then: make a right triangle.
Then: use a² + b² = c².
Last: check.