Download here

CP1 Math 2 Name_______________________________ Unit 5: Coordinate Geometry: Day 1 Coordinate Geometry: Using Slopes Recall: Finding Slope Slope =
𝐫𝐢𝐬𝐞
𝐫𝐮𝐧
slope=
𝒚𝟐 !𝒚𝟏
𝒙𝟐 !𝒙𝟏
Equations of Lines Slope-­‐intercept form: 𝑦 = 𝑚𝑥 + 𝑏 Point-­‐slope form: 𝑦 = 𝑚(𝑥 − ℎ) + 𝑘 Today we will use slopes to determine if two lines are parallel, perpendicular, or neither. Explore: Perpendicular Lines Each picture shows a pair of perpendicular lines. Find the slope of each line shown (there are 6 total). What do you notice? Generalize: How are the slopes of perpendicular lines related? Summary: Geometric Relation Algebraic Relation Parallel lines ∥ Lines that never intersect Slopes are equal Perpendicular lines ⊥ Lines that form right angles (90°) at their intersection Slopes are negative reciprocals (ex. ½ and -­‐2) Now we’ll use slope calculations to analyze special properties of some shapes. For example, we can use slope calculations to determine if a quadrilateral is a rectangle, parallelogram, trapezoid, or none of these things. Complete the following problems on a separate sheet of graph paper. 1. Consider ∆𝐴𝐵𝐶 with vertices A(6, 5), B(2, –1), and C(–10, 7). a. Draw the triangle on graph paper. b. Each side of the triangle is part of a line. Write an equation for each line. c. What is special about triangle ∆𝐴𝐵𝐶? Justify your answer using slopes. 2. You can calculate the midpoint of a line segment using averages. The midpoint’s x-­‐coordinate is the average of the x-­‐coordinates of the endpoints, and the midpoint’s y-­‐coordinate is the average of the y-­‐coordinates of the endpoints. a. Find the midpoint of the line segment from (–7, 5) to (3, 8). b. Generalize: Find the midpoint of the line segment from (x1, y1) to (x2, y2). 3. Refer back to ∆𝐴𝐵𝐶 from problem 1, where A(6, 5), B(2, –1), and C(–10, 7). a. Let 𝐷 be the midpoint of 𝐵𝐶, let 𝐸 be the midpoint of 𝐴𝐶, and let 𝐹 be the midpoint of 𝐴𝐵. Find the coordinates of 𝐷, of 𝐸 , and of 𝐹 . b. Add points 𝐷, 𝐸, and 𝐹 to your graph from problem 1. Use the graph to check that your answers to part a are correct. c. Connect 𝐷, 𝐸, and 𝐹 to form a new triangle, the “midpoint triangle.” Calculate the slopes of the three sides of the midpoint Δ𝐷𝐸𝐹. d. How do the slopes for the midpoint Δ𝐷𝐸𝐹compare to the slopes for ∆𝐴𝐵𝐶? Recall: •
A parallelogram is a quadrilateral (four-­‐sided shape) with two pairs of parallel sides. •
A trapezoid is a quadrilateral with only one pair of parallel sides. •
A rectangle is a quadrilateral with right angles at all four vertices (corners). 4. The lines with these equations form a quadrilateral: !
𝑦 = ! 𝑥 − 3 !
𝑦 = − ! 𝑥 + 6 !
𝑦 = ! (𝑥 − 2) + 4 !
𝑦 = ! (𝑥 + 3) − 1 a. Graph the lines on your calculator to help you visualize the quadrilateral. b. What are the slopes of the four sides? c. What special type of quadrilateral is it? Justify your answer using the definitions above. d. How many right angles does this quadrilateral have? Explain. 5. Consider quadrilateral 𝑊𝑋𝑌𝑍 with vertices W(–2, 3), X(–1, 8), Y(9, 6), and Z(8, 1). a. Is this quadrilateral a parallelogram? Justify your answer using slopes. b. Is this quadrilateral a trapezoid? Justify your answer using slopes. c. Is this quadrilateral a rectangle? Justify your answer using slopes. 6. Consider quadrilateral 𝑆𝑇𝑈𝑉 with vertices S(0, 2), T(2, 6), U(6, 4), and V(2, 1). a. Which angles of 𝑆𝑇𝑈𝑉 are right angles? Justify your answer using slopes. b. Prove that 𝑆𝑇𝑈𝑉 is a trapezoid. 7. Consider quadrilateral 𝑂𝑃𝑄𝑅 with O(2, 0), P(8, 4), Q(5, 8), and R(2, 6). a. Using slopes, determine whether quadrilateral 𝑂𝑃𝑄𝑅 is a parallelogram, a trapezoid, a rectangle, or none of these. b. Let 𝐾 , 𝐿, 𝑀, and 𝑁 stand for the midpoints of 𝑂𝑃, 𝑃𝑄, 𝑄𝑅, and 𝑅𝑂. Calculate the coordinates of these four points. c. Graph the quadrilateral and its midpoints on graph paper. Use your graph to check your answers to part b. d. Think of points 𝐾 , 𝐿, 𝑀, and 𝑁 as forming a new quadrilateral, the “midpoint quadrilateral.” Using slopes, determine whether quadrilateral 𝐾𝐿𝑀𝑁 is a parallelogram, a trapezoid, a rectangle, or none of these. Solutions (partial): Notation used throughout: mXY stands for the slope of the line through points X and Y. 1 and 3: D=(–4, 3), E=(–2,6), F=(4,2) mAB = mDE = 3/2 mAC = mDF = –1/8 mBC = mEF = –2/3 The slopes show that 𝐴𝐵 and 𝐵𝐶 are perpendicular, so ∆𝐴𝐵𝐶 is a right triangle. (Δ𝐷𝐸𝐹 is, too.) Each side of the Δ𝐷𝐸𝐹has the same slope as a side of ∆𝐴𝐵𝐶. ⎛ x + x 2 y1 + y 2 ⎞
2. In general, the midpoint is ⎜ 1
,
⎟ . 2 ⎠
⎝ 2
4. Trapezoid with two right angles 5. 𝑊𝑋𝑌𝑍 is both a parallelogram and a rectangle, because the slopes of the sides are mWX = mYZ = 5, mXY = mZW = –1/5. 6. 𝑆𝑇𝑈𝑉 is a trapezoid with two right angles, because the slopes of the sides are mST = 2, mTU = –1/2, mUV = 3/4, mVS = –1/2, which means that 𝑇𝑈 and 𝑉𝑆 are parallel and there are right angles at 𝑆 and 𝑇. 7. 𝑂𝑃𝑄𝑅 is a trapezoid because mOP = mQR = 2/3. K = (5, 2), L = (6.5, 6), M = (3.5, 7), N = (2, 3). 𝐾𝐿𝑀𝑁 is a parallelogram because mKN = mLM = –1/3, mKL = mMN = 4/1.5 or 8/3.