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Distances in Coordinate Geometry Presented for 10 th Grade Geometry class.
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Distances in Coordinate Geometry
Presented for 10th Grade Geometry class.
Distances in Coordinate Geometry
Objective:
Learn how to find the distance between two points on the coordinate plane.
Distances in Coordinate Geometry
Why do we need to know this?
There are many situations where we will need to measure a distance between two objects. What you are about to learn can make these problems easy!
Distances in Coordinate Geometry
We will use one of two methods:
• Pythagorean theorem
• Distance formula
PYTHAGOREAN THEOREM
C
B
A
In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.
Symbols: a2 + b2 = c2
ac
b
Homer: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”
Man in bathroom stall: “That's a right triangle, you idiot!”
Homer: “D'oh!”
The Wizard Of Oz, 1939 The Simpsons: “$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling)”
Distances using the Pythagorean Theorem
If we are given two points, then we can find the distance between them.
First, let’s connect the points with a line. Notice that the length of this line is exactly the distance we are trying to find.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
We want to measure the length of this line.
We need to draw 2 lines.
First, pick a point and draw a vertical line through it. 0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
We want to measure the length of this line.
Now draw a horizontal line through the other point.
The two lines that we drew will meet at a 90° angle.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
We want to measure the length of this line.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
We want to measure the length of this line.
We can use the Pythagorean theorem can help us.
If we want to use the Pythagorean Theorem to find c, then we need to know a and b.
Measuring, we find the lengths
a = 5b = 7
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
ca
b
Now use the Pythagorean theorem to find c.
a2 + b2 = c2
52 + 72 = c2
25 + 49 = c2
74 = c2
c is , or about 8.6 units long.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
c5
7
74
The distance between the two points is about 8.6 units.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
8.6
Distances using the Distance Formula
What we know
• We can find the distance between two points on a coordinate plane using a right triangle and the Pythagorean Theorem.
The Distance Formula
Let’s look again at how we solved the original problem.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
• First, we drew a right triangle.
• Then, we found the lengths of the two legs, a and b.
• Then we used the Pythagorean Theorem.
0
2
4
6
8
10
0 2 4 6 8 10
(2, 8)
(9, 3)
ca
b
The Distance Formula
0
2
4
6
8
10
0 2 4 6 8 10
a
b
c
The difference in x values, (x2 – x1) is the length of leg bIn this case, b = 9 – 2 = 7
The difference in y values, (y2 - y1) is the length of leg a.
In this case, a = (8 – 3) = 5
Notice how we can find the lengths of the legs a and b. We subtract the x-values and the y-values.
The Distance Formula
No matter what two points we are given, we will always subtract x and y values in this way to find a and b.
a = y2 – y1
b = x2 – x1
Note: It doesn’t matter in what order we subtract the numbers!
0
2
4
6
8
10
0 2 4 6 8 10
(x1, y1)
(x2, y2)
ca
b
The Distance Formula
Now, we need to use the Pythagorean Theorem to find the distance between the two points.
a2 + b2 = c2
(y2 – y1)2 + (x2 – x1)2 = c2
…but we want c, not c2. Do you remember how to do this?
0
2
4
6
8
10
0 2 4 6 8 10
(x1, y1)
(x2, y2)
ca
b
The Distance Formula
Done! Remember, c is the distance between the two points.
We have shown that
You should notice that we don’t need to draw anything when we use this formula. All we need to know is where the two points lie!
0
2
4
6
8
10
0 2 4 6 8 10
(x1, y1)
(x2, y2)
212
2
12 () yyxxc
The Distance Formula
Example of a Completed Problem
22 )83()29(
2
12
2
12 )()( yyxx
22 )5(7
In the figure given previously, the values of (x1, y1) and (x2, y2) are (2, 8) and (9, 3) respectively. Find the distance between the two points.
Distance =
Distance =
Distance =
Distance =
Distance =
742549
8.602
Distance Formula
Substitute Values
Evaluate Using Order
of Operations
Answer
The Distance Formula
