Distance Midpoint Distance Formula Pythagorean Theorem Midpoint Formula Fractional Location Midpoint

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: Simplify square roots Find the midpoint on a number line and in the coordinate plane Find the distance between two points on a number line and in the coordinate plane in simple radical form and rounded to nearest thousandth. Find the fractional distance between two points in the coordinate plane.

absolute value leg hypotenuse

District Formula (on Number Line) The distance between two points is the absolute value of the difference between their coordinates. If P has coordinates x1 and Q has coordinate x2. 𝑃𝑄= 𝑥 2 − 𝑥 1 or 𝑥 1 − 𝑥 2 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. 16.8 mm 50.4 mm A. 16.8 mm B. 57.4 mm C. 67.2 mm D. 84 mm

Use the number line to find AX. 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? 1 mile = 5280 feet A. 330 ft B. 660 ft C. 990 ft D. 1320 ft

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

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).

1 2 𝑥 1 + 𝑥 2 , 𝑦 1 + 𝑦 2

A. (–10, –6) B. (–5, –3) C. (6, 12) D. (–6, –12) 6 12

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 = 7 + y – 7 –7 – 2 –2 –5 = y 10 = x The coordinates of Y are (10, –5).

Let D be (x1, y1) and F be (x2, y2) in the Midpoint Formula. Write two equations to find the coordinates of D.

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

Two times the midpoint minus the other endpoint. S is the midpoint of 𝑹𝑻 . R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T. 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). B. (–10, 13) C. (15, –1) D. (17, –11)

( 𝑥 1 ± 𝑚 𝑥 2 − 𝑥 1 𝑛 , 𝑦 1 ± 𝑚 𝑦 2 − 𝑦 1 𝑛 ) Add to x1 if going right ( 𝑥 1 ± 𝑚 𝑥 2 − 𝑥 1 𝑛 , 𝑦 1 ± 𝑚 𝑦 2 − 𝑦 1 𝑛 ) B Add to x1 if going right Subtract from x1 if going left Add to y1 if going up Subtract from y1 if going down A

Find the coordinates of P, a point 1 4 of the distance from A(-2, -4) to B(4, 3). 𝟏 𝟒 AC = 𝟏 𝟒 (𝟔) = 𝟏.𝟓 −𝟐+𝟏.𝟓= −𝟎.𝟓 𝟏 𝟒 AC = 𝟏 𝟒 (𝟕) = 𝟕 𝟒 −𝟒+ 𝟕 𝟒 = −𝟐 𝟏 𝟒 A B C 7 { 3 2 7 4 } 1 4 {6, 7} + {-2, -4} { − 1 2 − 9 4 } 6 { −.5 −2.25}

7. Find P on NM that is the given fractional distance from N to M 7. Find P on NM that is the given fractional distance from N to M. Find 𝟏 𝟓 ; N(-3, -2), M(1, 1) −𝟑+ 𝟒 𝟓 = 𝟏 𝟓 MR = 𝟏 𝟓 (𝟒) = 𝟒 𝟓 −𝟐.𝟐 𝟏 𝟓 NR= 𝟏 𝟓 (𝟑) = 𝟑 𝟓 −𝟐+ 𝟑 𝟓 = −𝟏.𝟒 N 3 { .8 .6} M + {-3, -2} R .2 {4, 3} 4 { −2.2 −1.4}

(-2.2, -1.4) (0, -1.3) Round to nearest tenth (1, 2.3) (0.75, 4.75)

The Distance Formula & The Pythagorean Theorem

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

Find EF and GH. Then determine if 𝐄𝐅 ≅ 𝐆𝐇 . E(–2, 1), F(–5, 5), G(–1, –2), H(3, 1)

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

𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 𝟑+ 𝟒 𝟐 = 𝟓 𝟐 ? 25 u2 9 u2 16 u2

6 u2 6 u2 6 u2 6 u2

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. c2 = a2 + b2 = 52 + 92 = 25 + 81 = 106 c = 10.3

R(3, 2) and S(–3, –1) a = 6 and b = 3. c2 = a2 + b2 = 62 + 32 = 36 + 9 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) a = 6 and b = 3. c2 = a2 + b2 = 62 + 32 = 36 + 9 = 45

𝑑= (80−30) 2 + (5−15) 2 𝑑= (50) 2 + (−10) 2 𝑑= 2500+100 𝑑= 2600 Find the distance between (30, 15) and (80, 5). 𝑑= (80−30) 2 + (5−15) 2 𝑑= (50) 2 + (−10) 2 𝑑= 2500+100 𝑑= 2600 𝑑=10 26 𝑑=50.99

Given the points R(–4, 5) and S(2, –1) Given the points R(–4, 5) and S(2, –1). Find RS, the midpoint of 𝑅𝑆 and point P which is 1 3 of the distance from R to S. (Leave in simple radical form. 𝑅𝑆= (2−−4) 2 + (5−−1) 2 = 72 = 6 2 𝑅𝑆= (6) 2 + (6) 2 = 72 = 6 2 .5 −4 + 2, 5+−1 ={−1, 2} 1 3 CS (6) = 2 1 3 RC (6) = 2 –4 + 2 = –2 5 – 2 = 3 (–2 , 3)

Lesson Practice

1. Find the coordinates of the midpoint of 𝑀𝑁 with endpoints M(-2, 6) and N(8, 0). (3, 3) 2. K is the midpoint of 𝐻𝐿 . H has coordinates (1, –7), and K has coordinates (9, 3). Find the coordinates of L. (17, 13) 3. Find the distance, to the nearest tenth, between S(6, 5) and T(–3, –4). 12.7 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

5. Find the lengths of AB and CD and determine whether they are congruent.