Worked examples
The question
Question: find the distance between (1, 2) and (4, 6).
(1, 2) and (4, 6)
1 Plot the points
Draw the grid.
Put both points on the grid: (1, 2)
and (4, 6).
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 = 1 to x = 4 is 3.
We call it a = 3.
Then go up.
From y = 2 to y = 6 is 4.
We call it b = 4.
Across and up 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.
a2 + b2 = c2
Here c is the distance d.
c = d
Now put the numbers in: a = 3 and b = 4.
32 + 42 = d2
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: 9 + 16 = d².
9 + 16 = d2
9 plus 16 is 25.
25 = d2
d² is 25.
So d is the square root of 25.
d = √25
5 times 5 is 25.
5 × 5 = 25
So d = 5.
The distance is 5.
d = 5
4 Check
Let's check with the distance formula.
It does the same steps in one line.
d = √(x2 − x1)2 + (y2 − y1)2
Put in the numbers of (4, 6) and (1, 2).
= √(4 − 1)2 + (6 − 2)2
4 minus 1 is 3.
6 minus 2 is 4.
The same as on the grid.
= √32 + 42
9 plus 16 is 25.
The square root is 5 again.
Right!
= √25 = 5
Let's look back.
First: plot the points.
Then: make a right triangle.
Then: use a² + b² = c².
Last: check.
▶ Watch it solved
The question
Question: find the distance between (−2, 1) and (1, 5).
(−2, 1) and (1, 5)
1 Plot the points
Draw the grid.
Put both points on the grid: (−2, 1)
and (1, 5).
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 = 1 is 3.
We call it a = 3.
Then go up.
From y = 1 to y = 5 is 4.
We call it b = 4.
Across and up 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.
a2 + b2 = c2
Here c is the distance d.
c = d
Now put the numbers in: a = 3 and b = 4.
32 + 42 = d2
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: 9 + 16 = d².
9 + 16 = d2
9 plus 16 is 25.
25 = d2
d² is 25.
So d is the square root of 25.
d = √25
5 times 5 is 25.
5 × 5 = 25
So d = 5.
The distance is 5.
d = 5
4 Check
Let's check with the distance formula.
It does the same steps in one line.
d = √(x2 − x1)2 + (y2 − y1)2
Put in the numbers of (1, 5) and (−2, 1).
= √(1 − (−2))2 + (5 − 1)2
1 minus −2 is 3.
Taking away a negative is adding.
5 minus 1 is 4.
The same as on the grid.
= √32 + 42
9 plus 16 is 25.
The square root is 5 again.
Right!
= √25 = 5
Let's look back.
First: plot the points.
Then: make a right triangle.
Then: use a² + b² = c².
Last: check.
▶ Watch it solved
The question
Question: find the distance between (−4, −1) and (−1, −5).
(−4, −1) and (−1, −5)
1 Plot the points
Draw the grid.
Put both points on the grid: (−4, −1)
and (−1, −5).
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 = −4 to x = −1 is 3.
We call it a = 3.
Then go down.
From y = −1 to y = −5 is 4.
A length is never negative.
So b = 4.
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.
a2 + b2 = c2
Here c is the distance d.
c = d
Now put the numbers in: a = 3 and b = 4.
32 + 42 = d2
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: 9 + 16 = d².
9 + 16 = d2
9 plus 16 is 25.
25 = d2
d² is 25.
So d is the square root of 25.
d = √25
5 times 5 is 25.
5 × 5 = 25
So d = 5.
The distance is 5.
d = 5
4 Check
Let's check with the distance formula.
It does the same steps in one line.
d = √(x2 − x1)2 + (y2 − y1)2
Put in the numbers of (−1, −5) and (−4, −1).
= √(−1 − (−4))2 + (−5 − (−1))2
−1 minus −4 is 3.
Taking away a negative is adding.
−5 minus −1 is −4.
Taking away a negative is adding.
Careful: (−4)² is −4 times −4.
Minus times minus is plus: 16, not −16.
= √32 + (−4)2 (−4)² = 16, not −16
9 plus 16 is 25.
The square root is 5 again.
Right!
= √25 = 5
Let's look back.
First: plot the points.
Then: make a right triangle.
Then: use a² + b² = c².
Last: check.
▶ Watch it solved
