CONDENSED LESSON 11.1 Parallel and Perpendicular In this lesson you will ● learn the meaning of parallel and perpendicular ● discover how the slopes of parallel and perpendicular lines are related ● use slopes to help classify figures in the coordinate plane Parallel lines are lines in the same plane that never intersect. Perpendicular lines are lines in the same plane that intersect at a right angle. We show a small box in one of the angles to show that the lines are perpendicular. Investigation: Slopes The opposite sides of a rectangle are parallel, and the adjacent sides are perpendicular. By examining rectangles drawn on a coordinate grid, you can discover how the slopes of parallel and perpendicular lines are related. Step 1 gives the vertices of four rectangles. Here is the rectangle with y the vertices given in part a. A Find the slope of each side of the rectangle. You should get these 20 results. (Note: The notation AB៮ means “segment AB.”) 10 B 7 9 D Slope of AD៮៮: ᎏᎏ Slope of AB៮: Ϫᎏᎏ 9 7 x Ϫ 7 9 10 10 20 Slope of BC៮: ᎏᎏ Slope of DC៮៮: Ϫᎏᎏ C 9 7 Ϫ10 Notice that the slopes of the parallel sides AD៮៮ and BC៮ are the same and that the slopes of the parallel sides AB៮ and DC៮៮ are the same. Recall that, to find the reciprocal of a fraction, you exchange the numerator and the ᎏ3ᎏ ᎏ4ᎏ denominator. For example, the reciprocal of 4 is 3. The product of a number and its reciprocal is 1. Look at the slopes of the perpendicular sides AAD៮៮ and DC៮៮. The slope ៮៮ ៮៮ ᎏ7ᎏ of DC is the negative reciprocal of the slope of AD . The product of the slopes, 9 and Ϫᎏ9ᎏ Ϫ 7, is 1. You’ll find this same relationship for any pair of perpendicular sides of the rectangle. (continued) ©2002 Key Curriculum Press Discovering Algebra Condensed Lessons 143 Previous Next Lesson 11.1 • Parallel and Perpendicular (continued) Now, choose another set of vertices from Step 1, and find the slopes of the sides of the rectangle. You should find the same relationships among the slopes of the sides. In fact, any two parallel lines have the same slope, and any two perpendicular lines have slopes that are negative reciprocals. A right triangle has one right angle. The sides that form the right angle are called legs, and the side opposite the right angle is called the hypotenuse. If a Hypotenuse triangle is drawn on a coordinate grid, you can use what you know about slopes Leg of perpendicular lines to determine if it is a right triangle. This is demonstrated in Example A in your book. Here is another example. Leg EXAMPLE Decide whether this triangle is a right triangle. y 4 B 2 x Ϫ4 2 4 A Ϫ2 Ϫ 4 C ᮣ Solution This triangle has vertices A(Ϫ3, Ϫ2), B(Ϫ1, 2), and C(3, Ϫ4). Angles B and C are clearly not right angles, but angle A might be. To check, find the slopes of AB៮ and AC៮: 2 Ϫ (Ϫ2) 4 Ϫ4 Ϫ (Ϫ2) Ϫ2 1 Slope AB៮: ᎏᎏ ϭ ᎏᎏ ϭ 2 Slope AC៮: ᎏᎏ ϭ ᎏᎏ ϭϪᎏᎏ Ϫ1 Ϫ (Ϫ3) 2 3 Ϫ (Ϫ3) 6 3 Ϫᎏ1ᎏ The slopes, 2 and 3, are not negative reciprocals, so the sides are not perpendicular. Because none of the angles are right angles, the triangle is not a right triangle. A trapezoid is a quadrilateral with one pair of opposite sides that are parallel and one pair of opposite sides that are not parallel. A trapezoid with a right angle is a right trapezoid. Every right trapezoid must have two right angles because opposite sides are parallel. Here are some examples of trapezoids. To determine whether a quadrilateral drawn on a coordinate grid is a trapezoid that is not also a parallelogram, you need to check that two of the opposites sides have the same slope and the other two opposite sides have different slopes. To decide whether the trapezoid is a right trapezoid, you also need to check that the slopes of two adjacent sides are negative reciprocals. This is illustrated in Example B in your book. 144 Discovering Algebra Condensed Lessons ©2002 Key Curriculum Press Previous Next CONDENSED LESSON 11.2 Finding the Midpoint In this lesson you will ● discover the midpoint formula ● use the midpoint formula to find midpoints of segments ● write equations for medians of triangles and perpendicular bisectors of segments The midpoint of a line segment is the middle point—that is, the point halfway between the endpoints. The text on page 588 of your book explains that finding midpoints is necessary for drawing the median of a triangle and the perpendicular bisector of a line segment. Read this text carefully. Investigation: In the Middle This triangle has vertices A(1, 2), B(5, 2), and C(5, 7). y ៮ 8 The midpoint of AB is (3, 2). Notice that the x-coordinate of this point is C the average of the x-coordinates of the endpoints. 6 The midpoint of BC៮ is (5, 4.5). Notice that the y-coordinate of this point is the average of the y-coordinates of the endpoints. 4 ៮ The midpoint of AC is (3, 4.5). Notice that the x-coordinate of this 2 A B point is the average of the x-coordinates of the endpoints and that the y-coordinate is the average of the y-coordinates of the endpoints. x 0 2 4 6 Segment DE has endpoints D(2, 5) and E(7, 11). The midpoint y ៮ of DE is (4.5, 8). The x-coordinate of this point is the average of the 12 x-coordinates of the endpoints, and the y-coordinate is the average E of the y-coordinates of the endpoints. 10 Use the idea of averaging the coordinates of the endpoints to 8 Midpoint find the midpoint of the segment between each pair of points. (4.5, 8) For Step 8, you should get these results. 6 ៮ Ϫ D a. midpoint of FG :( 2.5, 28) 4 b. midpoint of HJ៮:(Ϫ1, Ϫ2) 2 x 0 2 4 6 8 The technique used in the investigation to find the midpoint of a segment is known as the midpoint formula. If the endpoints of a segment have coordinates (x1, y1) and (x2, y2), the midpoint of the segment has coordinates x ϩ x y ϩ y ᎏ1 ᎏ2 ᎏ1 ᎏ2 ΂ 2 , 2 ΃ (continued) ©2002 Key Curriculum Press Discovering Algebra Condensed Lessons 145 Previous Next Lesson 11.2 • Finding the Midpoint (continued) The example in your book shows how to find equations for a median of a triangle and the perpendicular bisector of one of its sides. Below is another example. EXAMPLE This triangle has vertices A(Ϫ2, 2), B(2, 4), and C(1, Ϫ3). y B 4 A 2 x Ϫ4 Ϫ2 2 4 Ϫ2 C Ϫ4 a. Write the equation of the median from vertex A. b. Write the equation of the perpendicular bisector of BC៮. ᮣ Solution a. The median from vertex A goes to the midpoint of BC៮. So, find the midpoint of BC៮. ϩ ϩ Ϫ ៮ ᎏ2 ᎏ1 ᎏ4 (ᎏ3) ϭ midpoint of BC : ΂ 2 , 2 ΃ (1.5, 0.5) Now, use the coordinates of vertex A and the midpoint to find the slope of the median. Ϫ Ϫ ᎏ0.5 ᎏ2 ϭ ᎏ1ᎏ.5 ϭϪᎏ3ᎏ slope of median: 1.5 Ϫ (Ϫ2) 3.5 7 Use the coordinates of the midpoint and the slope to find the equation. ϭ Ϫ ᎏ3ᎏ Ϫ y 0.5 7(x 1.5) b. The perpendicular bisector of BC៮ goes through the midpoint of BC៮, which is Ϫ Ϫ ៮ ៮ ᎏ3 ᎏ4 (1.5, 0.5) and is perpendicular to BC . The slope of BC is 1 Ϫ 2 , or 7, so the slope Ϫᎏ1ᎏ of the perpendicular bisector is the negative reciprocal of 7, or 7. Write the equation, using this slope and the coordinates of the midpoint. ϭ Ϫ ᎏ1ᎏ Ϫ y 0.5 7(x 1.5) 146 Discovering Algebra Condensed Lessons ©2002 Key Curriculum Press Previous Next CONDENSED LESSON 11.3 Squares, Right Triangles, and Areas In this lesson you will ● find the areas of polygons drawn on a grid ● find the area and side length of squares drawn on a grid ● draw a segment of a given length by drawing a square with the square of the length as area Example A in your book shows you how to find the length of a rectangle and a right triangle. Example B demonstrates how to find the area of a tilted square by drawing a square with horizontal and vertical sides around it. Read both examples carefully. Investigation:What’s My Area? Step 1 Find the area of each figure in Step 1. You should get these results. a. 1 square unit b. 5 square units c. 6 square units d. 2 square units e. 8 square units f. 3 square units g. 6 square units h. 6 square units i. 10.5 square units j. 8 square units There are many ways to find the areas of these figures. One useful technique involves drawing a rectangle around the figure. This drawing shows a rectangle around figure i. To find the area of the figure, subtract the sum of the areas of the triangles from the area of the rectangle. Area of figure i ϭ 3 и 6 Ϫ (2.5 ϩ 1 ϩ 2 ϩ 2) 2.5 1 ϭ 18 Ϫ 7.5 ϭ 10.5 2 2 Steps 2–4 If you know the area of a square, you can find the side length by taking the square root.