Structure question 2 doc

M = ( ᎏ –3 2 +1 ᎏ , ᎏ 5+ 2 (–4) ᎏ ) M = (– ᎏ 2 2 ᎏ , ᎏ 1 2 ᎏ ) M = (–1, ᎏ 1 2 ᎏ ) Note: There is no such thing as the midpoint of a line, as lines are infinite in length. SLOPE The slope of a line (or line segment) is a numerical value given to show how steep a line is. A line or segment can have one of four types of slope: ■ A line with a positive slope increases from the bottom left to the upper right on a graph. ■ A line with a negative slope decreases from the upper left to the bottom right on a graph. ■ A horizontal line is said to have a zero slope. ■ A vertical line is said to have no slope (undefined). ■ Parallel lines have equal slopes. ■ Perpendicular lines have slopes that are negative reciprocals of each other. Positive slope y 1 4 3 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 y 1 4 3 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 y 1 4 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 y 1 4 3 2 –5 –1 –2 –3 –4 1 542 –5 –1 –2 –3–4 x 5 3 3 Negative slope Zero slope Undefined (no) slope – THEA MATH REVIEW– 145 The slope of a line can be found if you know the coordinates of any two points that lie on the line. It does not matter which two points you use. It is found by writing the change in the y-coordinates of any two points on the line, over the change in the corresponding x-coordinates. (This is also known as the rise over the run.) The formula for the slope of a line (or line segment) containing points (x 1 , y 1 ) and (x 2 , y 2 ): m = ᎏ y x 2 2 – – y x 1 1 ᎏ . Example Determine the slope of the line joining points A(–3,5) and B(1,–4). Let (x 1 ,y 1 ) represent point A and let (x 2 ,y 2 ) represent point B. This means that x 1 = –3, y 1 = 5, x 2 = 1, and y 2 = –4. Substituting these values into the formula gives us: m = ᎏ x y 2 2 – – y x 1 1 ᎏ m = ᎏ 1 – – 4 ( – – 5 3) ᎏ m = ᎏ – 4 9 ᎏ Example Determine the slope of the line graphed below. Two points that can be easily determined on the graph are (3,1) and (0,–1). Let (3,1) = (x 1 , y 1 ), and let (0,–1) = (x 2 , y 2 ). This means that x 1 = 3, y 1 = 1, x 2 = 0, and y 2 = –1. Substituting these values into the formula gives us: y 1 4 3 2 –5 –1 –2 –3 –4 1 5 4 32 –5 –1–2–3–4 x 5 – THEA MATH REVIEW– 146 m = ᎏ – 0 1 – – 3 1 ᎏ m = ᎏ – – 2 3 ᎏ = ᎏ 2 3 ᎏ Note: If you know the slope and at least one point on a line, you can find the coordinate point of other points on the line. Simply move the required units determined by the slope. For example, from (8,9), given the slope ᎏ 7 5 ᎏ , move up seven units and to the right five units. Another point on the line, thus, is (13,16). Determining the Equation of a Line The equation of a line is given by y = mx + b where: ■ y and x are variables such that every coordinate pair (x,y) is on the line ■ m is the slope of the line ■ b is the y-intercept, the y-value at which the line intersects (or intercepts) the y-axis In order to determine the equation of a line from a graph, determine the slope and y-intercept and substi- tute it in the appropriate place in the general form of the equation. Example Determine the equation of the line in the graph below. y 4 2 –2 –4 4 2 –2–4 x – THEA MATH REVIEW– 147 In order to determine the slope of the line, choose two points that can be easily determined on the graph. Two easy points are (–1,4) and (1,–4). Let (–1,4) = (x 1 , y 1 ), and let (1,–4) = (x 2 , y 2 ). This means that x 1 = –1, y 1 = 4, x 2 = 1, and y 2 = –4. Substituting these values into the formula gives us: m = ᎏ 1 – – 4 ( – – 4 1) ᎏ = ᎏ – 2 8 ᎏ = – 4. Looking at the graph, we can see that the line crosses the y-axis at the point (0,0). The y-coordinate of this point is 0. This is the y-intercept. Substituting these values into the general formula gives us y = –4x + 0, or just y = –4x. Example Determine the equation of the line in the graph below. Two points that can be easily determined on the graph are (–3,2) and (3,6). Let (–3,2) = (x 1 ,y 1 ), and let (3,6) = (x 2 ,y 2 ). Substituting these values into the formula gives us: m = ᎏ 3 6 – – (– 2 3) ᎏ = ᎏ 4 6 ᎏ = ᎏ 2 3 ᎏ . We can see from the graph that the line crosses the y-axis at the point (0,4). This means the y-intercept is 4. Substituting these values into the general formula gives us y = ᎏ 2 3 ᎏ x + 4. y 4 2 –2 –4 42 –2–4 x 6 –6 –6 6 – THEA MATH REVIEW– 148 Angles NAMING ANGLES An angle is a figure composed of two rays or line segments joined at their endpoints. The point at which the rays or line segments meet is called the vertex of the angle. Angles are usually named by three capital letters, where the first and last letter are points on the end of the rays, and the middle letter is the vertex. This angle can either be named either ∠ABC or ∠CBA, but because the vertex of the angle is point B,letter B must be in the middle. We can sometimes name an angle by its vertex, as long as there is no ambiguity in the diagram. For exam- ple, in the angle above, we may call the angle ∠B, because there is only one angle in the diagram that has B as its vertex. But, in the following diagram, there are a number of angles which have point B as their vertex, so we must name each angle in the diagram with three letters. Angles may also be numbered (not measured) with numbers written between the sides of the angles, on the interior of the angle, near the vertex. CLASSIFYING ANGLES The unit of measure for angles is the degree. Angles can be classified into the following categories: acute, right, obtuse, and straight. 1 B C A F D E G B C A – THEA MATH REVIEW– 149 ■ An acute angle is an angle that measures between 0 and 90 degrees. ■ A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90°, but less than 180°. ■ A straight angle is an angle that measures 180°. Thus, both of its sides form a line. Straight Angle 180° Obtuse Angle Right Angle Symbol A cute Angle – THEA MATH REVIEW– 150 . slope y 1 4 3 2 –5 –1 2 –3 –4 1 54 32 –5 –1 2 –3–4 x 5 y 1 4 3 2 –5 –1 2 –3 –4 1 54 32 –5 –1 2 –3–4 x 5 y 1 4 2 –5 –1 2 –3 –4 1 54 32 –5 –1 2 –3–4 x 5 y 1 4 3 2 –5 –1 2 –3 –4 1 5 42 –5 –1 2 –3–4 x 5 3 3 Negative. and let (0,–1) = (x 2 , y 2 ). This means that x 1 = 3, y 1 = 1, x 2 = 0, and y 2 = –1. Substituting these values into the formula gives us: y 1 4 3 2 –5 –1 2 –3 –4 1 5 4 32 –5 –1 2 3–4 x 5 – THEA. y 1 ) and (x 2 , y 2 ): m = ᎏ y x 2 2 – – y x 1 1 ᎏ . Example Determine the slope of the line joining points A(–3,5) and B(1,–4). Let (x 1 ,y 1 ) represent point A and let (x 2 ,y 2 ) represent