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 B A C 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

Man in bathroom stall: “That's a right triangle, you idiot!” The Wizard Of Oz, 1939 The Simpsons: “$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling)” 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!”

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. 10 (2, 8) We want to measure the length of this line. 8 6 4 (9, 3) 2 2 4 6 8 10

First, pick a point and draw a vertical line through it. We need to draw 2 lines. First, pick a point and draw a vertical line through it. 10 (2, 8) We want to measure the length of this line. 8 6 4 (9, 3) 2 2 4 6 8 10

Now draw a horizontal line through the other point. The two lines that we drew will meet at a 90° angle. 10 (2, 8) We want to measure the length of this line. 8 6 4 (9, 3) 2 2 4 6 8 10

We can use the Pythagorean theorem can help us. 10 (2, 8) We can use the Pythagorean theorem can help us. We want to measure the length of this line. 8 6 4 (9, 3) 2 2 4 6 8 10

Measuring, we find the lengths a = 5 b = 7 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 = 5 b = 7 10 (2, 8) 8 c 6 a 4 b (9, 3) 2 2 4 6 8 10

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. 10 (2, 8) 8 c 6 5 4 7 (9, 3) 2 2 4 6 8 10

The distance between the two points is about 8.6 units. 10 (2, 8) The distance between the two points is about 8.6 units. 8 6 8.6 4 (9, 3) 2 2 4 6 8 10

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 10 (2, 8) Let’s look again at how we solved the original problem. 8 6 4 (9, 3) 2 2 4 6 8 10

The Distance Formula First, we drew a right triangle. Then, we found the lengths of the two legs, a and b. Then we used the Pythagorean Theorem. 10 (2, 8) 8 c 6 a 4 b (9, 3) 2 2 4 6 8 10

The Distance Formula Notice how we can find the lengths of the legs a and b. We subtract the x-values and the y-values. 10 The difference in y values, (y2 - y1) is the length of leg a. In this case, a = (8 – 3) = 5 8 c 6 a 4 b 2 2 4 6 8 10 The difference in x values, (x2 – x1) is the length of leg b In this case, b = 9 – 2 = 7

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! 10 (x1, y1) 8 c 6 a 4 b (x2, y2) 2 2 4 6 8 10

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? 10 (x1, y1) 8 c 6 a 4 b (x2, y2) 2 2 4 6 8 10

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! 10 (x1, y1) 8 6 4 (x2, y2) 2 2 4 6 8 10

The Distance Formula Example of a Completed Problem 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 Formula Substitute Values Evaluate Using Order of Operations Answer 8.602