1-6 Midpoint and distance in the coordinate plane Chapter 1 1-6 Midpoint and distance in the coordinate plane

Objectives Students will be able to : Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points.

Midpoint Formula What is the midpoint ? Answer: It is the middle section of a line. Is what divides a line in 2 congruent sections. How can you find the midpoint of a line? You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Midpoint Formula

Example 1 Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7). Solution: 𝑚 𝑥1+𝑥2 2 , 𝑦1+𝑦2 2 midpoint formula 𝑚 −8+−2 2 , 3+7 2 𝑚 − 10 2 , 10 2 𝑚[−5,5]

Example 2 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3). Solution: 𝑚 𝑥1+𝑥2 2 , 𝑦1+𝑦2 2 midpoint formula 𝑚 −2+5 2 , 3+ −3 2 𝑚 3 2 , 0 2 𝑚[ 3 2 ,0]

Student guide Do problems 2 and3 from page 47 from book

Example 3 Lets do problems 1 and 2 from worksheet

Student Guide Lets do problems 3 to 5 from worksheet

Example 4 M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Solution:

Student guided practice Lets do problems 4 to 5 from book page 47 Worksheet problems 21-23

Distance formula What is the distance formula? . The Distance Formula is used to calculate the distance between two points in a coordinate plane.

Example 5 Lets do distance formula worksheet

Pythagorean theorem What is the Pythagorean theorem? Answer: Is the theorem we used in right triangles to figure out a side of the triangle. You can also used it to find the distance between two points in the coordinate plane. In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c.

Example 6 Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R to S. R(3, 2) and S(–3, –1)

Example 6 continue Method 1 Use the Distance Formula. Substitute the values for the coordinates of R and S into the Distance Formula.

Example 6 continue Method 2 Use the Pythagorean Theorem. Count the units for sides a and b.

Example 7 DO problems 1-2 in worksheet

Student guided practice Let students do Pythagorean workshweet

Pythagorean A player throws the ball from first base to a point located between third base and home plate and 10 feet from third base. What is the distance of the throw, to the nearest tenth?

Homework DO problems 8-15 in the book page 47

Closure Today we saw the midpoint formula, the distance formula and the Pythagorean Theorem.