Step by step
The question
Question: find the distance between (2, 1) and (8, 9).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Plot the points
Draw the grid.
Put both points on the grid: (2, 1)
and (8, 9).
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 = 8.
How far?
A length is never negative, so take the smaller x away from the bigger one.
8 minus 2 is 6.
We call it a = 6.
Then go up: from y = 1 to y = 9.
Again, the bigger y take away the smaller one.
9 minus 1 is 8.
We call it b = 8.
Across and up make a right angle.
The distance d is the longest side, across from the right angle.
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 = 6 and b = 8.
6² means 6 times 6.
That is 36.
8² means 8 times 8.
That is 64.
Write the squares: 36 + 64 = d².
36 plus 64 is 100.
d² is 100.
So d times d is 100.
That means d is the square root of 100.
10 times 10 is 100.
So d = 10.
The distance is 10.
4 Check
Let's check with the distance formula.
It does the same steps in one line.
Under the root: the across squared, plus the up squared.
Put in the numbers: x₂ and y₂ from (8, 9), x₁ and y₁ from (2, 1).
8 minus 2 is 6.
9 minus 1 is 8.
The same as on the grid.
36 plus 64 is 100.
The square root is 10 again.
Right!
5 Pick the choice
Our answer is d = 10.
That is choice A.
6 Why not the others
B is 10√2.
Wrong: added the coordinates instead of taking them away.
C is 14.
Wrong: 6 + 8: added across and up; the distance is the slanted side.
D is 100.
Wrong: 36 + 64 = 100, but we still had to take the square root.
Let's look back.
First: plot the points.
Then: make a right triangle.
Then: use a² + b² = c².
Then: check.
Then: pick the choice.
Last: why not the others.