Distance Distance Formula Pythagorean Theorem Midpoint Midpoint Formula

Distance • Distance Formula • Pythagorean Theorem Midpoint • Midpoint Formula • Fractional Location

You Will: • Find the distance & midpoint between two point on a number line. • Find the distance between two points in a coordinate plane using the Distance Formula and the Pythagorean Theorem. • Find the midpoint of a segment. • Locate a point on a segment given a fractional distance from one endpoint.

Success Criteria: 1) Simplify square roots 2) Find the midpoint on a number line and in the coordinate plane. 3) Find the other endpoint of a segment given the midpoint and one endpoint 4) Find the distance between two points on a number line and in the coordinate plane in simple radical form and rounded to nearest thousandth. 5) Find the fractional distance between two points in the coordinate plane.

District Formula (on Number Line) The distance between two points is the absolute value of the difference between their coordinates. Use the number line to find QR. The coordinates of Q and R are – 6 and – 3. QR = | – 6 – (– 3) | = | – 3 | or 3

Use the number line to find each measure.

Find BD. Assume that the figure is not drawn to scale. A. 16. 8 mm D 50. 4 mm B. 57. 4 mm C. 67. 2 mm 16. 8 mm B D. 84 mm C

Use the number line to find AX. A. 2 B. 8 C. – 2 D. – 8

The length of a drag racing strip is ¼ mile long; How many feet from the finish line is the midpoint of the racing strip? A. 330 ft B. 660 ft C. 990 ft D. 1320 ft 1 mile = 5280 feet

Helpful Hint To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane.

Find the coordinates of the midpoint of PQ with endpoints P(– 8, 3) and Q(– 2, 7). = (– 5, 5)

Find the coordinates of the midpoint of EF with endpoints E(– 2, 3) and F(5, – 3).

M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y). Step 2 Use the Midpoint Formula: 12 = 2 + x – 2 10 = x 2=7+y – 7 – 5 = y The coordinates of Y are (10, – 5).

M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. The coordinates of Y are (10, – 5).

M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. The coordinates of Y are (10, – 5).

Let D be (x 1, y 1) and F be (x 2, y 2) in the Midpoint Formula. (x 2, y 2) = (– 5, – 3) Write two equations to find the coordinates of D.

Midpoint Formula Answer: The coordinates of D are (– 7, 11).

Two times the midpoint minus the other endpoint.

Find the coordinates of R if N (8, – 3) is the midpoint of RS and S has coordinates (– 1, 5). A. (3. 5, 1) B. (– 10, 13) C. (15, – 1) D. (17, – 11)

B • Add to x 1 if going right • Subtract from x 1 if going left • Add to y 1 if going up • Subtract from y 1 if going down A

Method 2 B 7 A C 6

M 3 N 4 R

M 8 (0, – 1. 3) N 6 R

8 N 1 M R (1, 2. 3) 12

M 5 N 5 (0. 75, 4. 75) R

Round to nearest tenth (-2. 2, -1. 4) (1, 2. 3) (0, -1. 3) (0. 75, 4. 75)

Locating a Point Given a Ratio (– 4, 6) 3. 6 4 C F(– 1, 3. 6) F 5

The Distance Formula & The Pythagorean Theorem

Helpful Hint To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane.

You can also use the Pythagorean Theorem to find the distance between two points in a 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.

What do you call a tea pot used on Mt. Everest to make tea? Hypotenuse High pot in use

? 25 u 2 9 u 2 16 u 2

6 u 2

Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(– 2, – 5). Half the class will use the Distance Formula the other half will use the Pythagorean Theorem.

Method 1 Use the Distance Formula. Substitute the values for the coordinates of D and E into the Distance Formula. D(3, 4) to E(– 2, – 5)

Method 2 Use the Pythagorean Theorem. Count the units for sides a and b. a = 5 and b = 9. c 2 = a 2 + b 2 = 52 + 9 2 = 25 + 81 = 106 c = 10. 3

Use the Distance Formula and the Pythagorean Theorem to find the distance, from R to S. Leave in simple radical form. R(3, 2) and S(– 3, – 1) a = 6 and b = 3. c 2 = a 2 + b 2 = 62 + 3 2 = 36 + 9 = 45

Find the distance between (30, 15) and (80, 5).

On the back of your notes – 4 + 2 = – 2 5– 2=3 (– 2 , 3)

Q: What happen to the plant in the math classroom? A. It grew square roots.

Lesson Practice

(3, 3) (17, 13) 3. Find the distance, between S(6, 5) and T(– 3, – 4). Leave in simple radical form. 4. The coordinates of the vertices of ∆ABC are A(2, 5), B(6, – 1), and C(– 4, – 2). Find the perimeter of ∆ABC, to the nearest tenth. 26. 5