LESSON 10.1 Parallel and Perpendicular

1 2008 Key Curriculum PressDiscovering Algebra Condensed Lessons131 Parallel and PerpendicularIn this LESSON you will learn the meaning ofparallelandperpendicular discover how the slopes of Parallel and Perpendicular lines are related use slopes to help classify figuresin the coordinate plane learn about inductiveanddeductive reasoningParallel linesare lines in the same plane that never intersect. Notice the marks thatindicate Parallel linesare lines in the same plane that intersect at a right angle,which measures 90 . You draw a small box in one of the angles to show that the linesare : SlopesThe opposite sides of a rectangle are Parallel , and the adjacent sides areperpendicular.

2 By examining rectangles drawn on a coordinate grid, you candiscover how the slopes of Parallel and Perpendicular lines are 1 gives the vertices of four rectangles. Here is the rectangle with the vertices given in part the slope of each side of the rectangle. You should get theseresults. (Note:The notation AB means segment AB. )Slope ofAD : 79 Slope ofAB : 97 Slope ofBC : 79 Slope ofDC : 97 Notice that the slopes of the Parallel sides AD andBC are the sameand that the slopes of the Parallel sides AB andDC are the that, to find the reciprocalof a fraction, you exchange the numerator and thedenominator.

3 For example, the reciprocal of 34 is 43 . The product of a number and itsreciprocal is 1. Look at the slopes of the Perpendicular sides AAD andDC . The slopeofDC is the opposite reciprocalof the slope ofAD . The product of the slopes, 79 and 97 ,is 1. You ll find this same relationship for any pair of Perpendicular sides ofthe , choose another set of vertices from Step 1, and find the slopes of the sides ofthe rectangle. You should find the same relationships among the slopes of the fact, any two Parallel lines have the same slope, and any two Perpendicular lineshave slopes that are opposite 10 101020 ABCD(continued) Algebra Condensed Lessons 2008 Key Curriculum PressLesson Parallel and Perpendicular (continued)Aright trianglehas one right angle.

4 The sides that form the right angle arecalledlegs,and the side opposite the right angle is called the atriangle is drawn on a coordinate grid, you can use what you know about slopesof Perpendicular lines to determine whether it is a right triangle. This isdemonstrated in Example A in your book. Here is another whether this triangle is a right triangle. SolutionThis triangle has vertices A( 3, 2),B( 1, 2), and C(3, 4). Angles BandCare clearly not right angles, but angle Amight be. To check, find the slopes ofAB andAC :Slope AB : 21 ( ( 23)) 42 2 Slope AC : 34 ( ( 32)) 62 13 The slopes, 2 and 13 , are not opposite reciprocals, so the sides are not none of the angles are right angles, the triangle is not a right read about inductive reasoninganddeductive reasoningon page a quadrilateral with one pair of opposite sides that are Parallel andone pair of opposite sides that are not Parallel .

5 A trapezoid with a right angle is aright right trapezoid must have two right angles because oppositesides are Parallel . Here are some examples of 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 sideshave 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 theslopes of two adjacent sides are negative reciprocals. This is illustrated in Example Bin your about several additional types of special quadrilaterals on page 555 ofyour 4 2 442 CABLegHypotenuseLeg 2008 Key Curriculum PressDiscovering Algebra Condensed Lessons133 Finding the this LESSON you will discover the midpoint formula use the midpoint formula to find midpoints of segments write equations for mediansof triangles and Perpendicular bisectorsof segmentsThemidpointof a line segment is the middle point that is, the point halfwaybetween the endpoints.

6 The text on page 557 of your book explains thatfinding midpoints is necessary for drawing the medianof a triangle and theperpendicular bisectorof a line segment. Read this text : In the MiddleThis triangle has vertices A(1, 2),B(5, 2), and C(5, 7).The midpoint ofAB is (3, 2). Notice that the x-coordinate of this point isthe average of the x-coordinates of the midpoint ofBC is (5, ). Notice that the y-coordinate of this pointis the average of the y-coordinates of the midpoint ofAC is (3, ). Notice that the x-coordinate of thispoint is the average of the x-coordinates of the endpoints and that they-coordinate is the average of the y-coordinates of the DEhas endpoints D(2, 5) and E(7, 11).

7 The midpoint ofDE is ( , 8). The x-coordinate of this point is the average of thex-coordinates of the endpoints, and the y-coordinate is the averageof the y-coordinates of the the idea of averaging the coordinates of the endpoints tofind the midpoint of the segment between each pair of Step 7, you should get these ofFG :( , 28) ofHJ :( 1, 2)The midpoint of the segment connecting (a,b) and (c,d) is a 2c , b 2d .The technique used in the investigation to find the midpoint of a segment is knownas the midpoint the endpoints of a segment have coordinates x1,y1 and x2,y2 , the midpoint of the segment has coordinates x1 2x2 , y1 2y2 x420y8624681012 EDMidpoint( , 8)xy42062468 ABC(continued)134 Discovering Algebra Condensed Lessons 2008 Key Curriculum PressLesson Finding the Midpoint (continued)The example in your book shows how to find equations for a median of a triangleand the Perpendicular bisector of one of its sides.

8 Here is another triangle has vertices A( 2, 2),B(2, 4), and C(1, 3). the equation of the median from vertex the equation of the Perpendicular bisector ofBC . median from vertex Agoes to the midpoint ofBC . So, find the midpoint ofBC .midpoint ofBC : 2 21 , 4 2( 3) ( , )Now, use the coordinates of vertex Aand the midpoint to find the slope ofthe of median: 5 ( 22) 37 Use the coordinates of the midpoint and the slope to find the 37 (x ) Perpendicular bisector ofBC goes through the midpoint ofBC ,which is( , ), and is Perpendicular to BC.

9 The slope ofBC is 13 24 , or 7, so theslope of the Perpendicular bisector is the opposite reciprocal of 7, or 17 .Writethe equation, using this slope and the coordinates of the 17 (x )xy42 4 2 4 242 ACB 2008 Key Curriculum PressDiscovering Algebra Condensed Lessons135 Squares, Right Triangles, and this LESSON you will find the areas of polygonsdrawn on a grid find the areaandside length of squares drawn on a grid draw a segment of a given length by drawing a square with the square of thelength as areaExample A in your book shows you how to find the area of a rectangle and a righttriangle.

10 Example B demonstrates how to find the area of a tilted square by drawinga square with horizontal and vertical sides around it. Read both examples : What s My Area?Step 1 Find the area of each figure in Step 1. You should get these square square square square square square square square square square unitsThere are many ways to find the areas of these figures. One useful technique involvesdrawing a rectangle around the figure. This drawing shows a rectangle aroundfigure i. To find the area of the figure, subtract the areas of the triangles from thearea of the of figure i 3 6 ( 1 2 2) 18 2 4If you know the area of a square, you can find the side length by takingthe square root.