Midpoint and Distance Formulas

Transcription

1 CP1 Math Unit 5: Coordinate Geometry: Day Name Midpoint Formula: Midpoint and Distance Formulas The midpoint of the line segment between any two points (x!, y! ) to (x!, y! ) is given by: In your groups, explain why the midpoint formula works. Exploration: Distance Formula B 4 A F 1. Find the distance between point A and point B. Which coordinates are useful here (x or y)? C 5 5 D. Find the distance between point C and point D. Which coordinates are useful here (x or y)? E 4 3. Find the distance between point E and point F by following the process below. a. Draw horizontal and vertical segments to create a right triangle. b. Find the coordinates of the third point in the right triangle, G. c. Find the lengths of EG and FG. d. Use the lengths of EG and FG to find the exact length of EF.

2 4. What if the two points are so far apart that it is not practical to plot them? Generalize the distance between any two points (x!, y! ) and (x!, y! ). (x, y ) d (x 1, y 1 ) ( ) The Pythagorean Theorem tells us that Square root this equation to get the Distance Formula: Note: The proof just stated does not apply to every pair of points, because sometimes the diagram for two points will look different from the one above. The Distance Formula still applies in these other cases, but the proof would need to be adjusted. This issue is partly addressed in problem 5. Problem Set: Complete on a separate sheet of graph paper. 1. Given the points ( 4, 7) and (, 5), find each of the following: a. the distance between the points b. the midpoint of the points c. the slope of the line through the points d. an equation for the line through the points

3 . Here is the diagram from problems 1 and 3 in the last problem set. You can refer back to those problems to get the coordinates of the points. Points D, E, and F are midpoints of the sides of the large triangle. a. Use the distance formula to find the lengths AB and DE. How are these lengths related? (If you don t see a relationship, make sure to simplify the square roots) b. Repeat for AC compared to DF, and for BC compared to EF. c. Describe in words what you ve observed about lengths in this diagram. 3. A quadrilateral whose four side lengths are all equal is called a rhombus. Determine whether each set of points forms a rhombus. Drawing the quadrilaterals on graph paper will help. a. (0, 1), (3, 5), ( 4, ), and ( 1, ). b. ( 3, 3), (1, 6), (, 1), and (, ). c. (, 1), (3, 3), (8, 1), and (3, 1). d. (, ), (4, 5), (1, 7), and ( 1, 4). 4. A quadrilateral that is both a rectangle and a rhombus (has both equal angles and equal sides) is called a square. Which of the quadrilaterals from problem 3 is a square? Prove your answer using distances and slopes.

4 5. The Distance Formula proof on page 1 did not account for the possibility that the points could be located differently in relation to each other. The diagram at the right shows one of the other possible arrangements of the points. a. The length of the horizontal leg in this triangle is not x x 1. What is the length? b. Does the formula d = ( x x1 ) + ( y y1) still apply for this diagram? If yes, explain why. If no, explain how the formula must be changed. (x, y) d (x1, y1) 6. Is the triangle shown below an equilateral triangle, an isosceles triangle, or a scalene triangle? Justify your answer using coordinate calculations. 7. Consider quadrilateral EFGH with E = (1, 1), F = (4, 5), G = (0, ), and H = ( 3, ). a. Is EFGH a rectangle? Justify your answer. b. Is EFGH a rhombus? Justify your answer. c. Is EFGH a square? Justify your answer. d. Consider the diagonals EG and FH. For each diagonal, find the length and the slope. e. Use the results of part d to justify EG FH f. Which of the following is the best explanation of why the diagonals turned out to be perpendicular? I. Rectangles always have perpendicular diagonals. II. Rhombuses always have perpendicular diagonals. III. Only squares have perpendicular diagonals.

5 8. Below is the triangle from problem again. Let D, E, and F be the midpoints of the sides. Remember from our first Geometry unit that a line or line segment from the vertex of a triangle to the midpoint of the opposite side is called a median. a. Find the coordinates of D, E, and F. b. Find an equation for AD. c. Find an equation for BE. d. Find an equation for CF. e. Using the equations from b, c, and d, show that point (0, 1) lies on all three medians.

6 Select answers 1 1. a. 40 = 10 b. ( 1, 6) c. 3 d. y = 1 3 ( x + 4) + 7 or y = 1 3 ( x ) + 5 both equal y = 1 3. a. AB = 5 = 13, DE = 13 Length DE is half of length AB. b. AC = 60 = 65, DF = 65, so DF is half of AC. BC = 08 = 4 13, EF = 13, so EF is half of BC. x c. Each midpoint-to-midpoint segment has half the length of the side of the original triangle that is parallel to it. 3. a. rhombus, all side lengths are 5 = 5 b. not a rhombus, side lengths vary c. rhombus, all side lengths are 9 d. rhombus, all side lengths are The quadrilateral from 3d is a square. At each vertex, the slopes that come together are /3 and 3/ which are opposite reciprocals. Therefore all the angles are right angles, which means all the angles are equal. Since the quadrilateral has both equal sides and equal angles, it is a square. 5. The length is (x 1 x ) rather than (x x 1 ), but the formula for d remains valid because (x 1 x ) = (x x 1 ). (The order of the subtraction doesn t matter because there s squaring afterward, and opposite numbers have equal squares.) 6. isosceles triangle because the side lengths are 90, 90, and 7 7. rhombus with perpendicular diagonals (slopes of 1 and 1) 8. Easiest to use point-slope form for parts b, c, and d. For example, y = 1(x 4) 3 is a correct answer for b. For part e, take each equation, put in x = 0 and y = 1, and show the equation is true. If all are true, then you know your answers to b/c/d are correct.