Theatex as higher education assessment_6 pot

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 SPECIAL ANGLE PAIRS ■ Adjacent angles are two angles that share a common vertex and a common side. There is no numerical relationship between the measures of the angles. ■ A linear pair is a pair of adjacent angles whose measures add to 180°. ■ Supplementary angles are any two angles whose sum is 180°. A linear pair is a special case of supplemen- tary angles. A linear pair is always supplementary, but supplementary angles do not have to form a linear pair. ■ Complementary angles are two angles whose sum measures 90 degrees. Complementary angles may or may not be adjacent. Example Two complementary angles have measures 2x° and 3x + 20°. What are the measures of the angles? Since the angles are complementary, their sum is 90°. We can set up an equation to let us solve for x: 2x + 3x + 20 = 90 5x + 20 = 90 5x = 70 x = 14 Substituting x = 14 into the measures of the two angles, we get 2(14) = 28° and 3(14) + 20 = 62°. We can check our answers by observing that 28 + 62 = 90, so the angles are indeed complementary. 50 ˚ 40 ˚ 50 ˚ Adjacent complementary angles Non-adjacent complementary angles 40 ˚ 70 ˚ 110 ˚ 70 ˚ 110 ˚ Linear pair (also supplementary) Supplementary angles (but not a linear pair) 1 2 1 2 Adjacent angles ∠1 and ∠2 Non-adjacent angles ∠1 and ∠2 – THEA MATH REVIEW– 151 Example One angle is 40 more than 6 times its supplement. What are the measures of the angles? Let x = one angle. Let 6x + 40 = its supplement. Since the angles are supplementary, their sum is 180°. We can set up an equation to let us solve for x: x + 6x + 40 = 180 7x + 40 = 180 7x = 140 x = 20 Substituting x = 20 into the measures of the two angles, we see that one of the angles is 20° and its supplement is 6(20) + 40 = 160°. We can check our answers by observing that 20 + 160 = 180, prov- ing that the angles are supplementary. Note: A good way to remember the difference between supplementary and complementary angles is that the letter c comes before s in the alphabet; likewise “90” comes before “180” numerically. ANGLES OF INTERSECTING LINES Important mathematical relationships between angles are formed when lines intersect. When two lines intersect, four smaller angles are formed. Any two adjacent angles formed when two lines intersect form a linear pair, therefore they are supplemen- tary. In this diagram, ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, and ∠4 and ∠1 are all examples of linear pairs. Also, the angles that are opposite each other are called vertical angles. Vertical angles are angles who share a vertex and whose sides are two pairs of opposite rays. Vertical angles are congruent. In this diagram, ∠1 and ∠3 are vertical angles, so ∠1 ≅∠3; ∠2 and ∠4 are congruent vertical angles as well. Note: Vertical angles is a name given to a special angle pair. Try not to confuse this with right angle or per- pendicular angles, which often have vertical components. 2 1 3 4 – THEA MATH REVIEW– 152 Example Determine the value of y in the diagram below: The angles marked 3y + 5 and 5y are vertical angles, so they are congruent and their measures are equal. We can set up and solve the following equation for y: 3y + 5 = 5y 5 = 2y 2.5 = y Replacing y with the value 2.5 gives us the 3(2.5) + 5 = 12.5 and 5(2.5) = 12.5. This proves that the two vertical angles are congruent, with each measuring 12.5°. PARALLEL LINES AND TRANSVERSALS Important mathematical relationships are formed when two parallel lines are intersected by a third, non-parallel line called a transversal. In the diagram above, parallel lines l and m are intersected by transversal n. Supplementary angle pairs and vertical angle pairs are formed in this diagram, too. Supplementary Angle Pairs Vertical Angle Pairs ∠1 and ∠2 ∠2 and ∠4 ∠1 and ∠4 ∠4 and ∠3 ∠3 and ∠1 ∠2 and ∠3 ∠5 and ∠6 ∠6 and ∠8 ∠5 and ∠8 ∠8 and ∠7 ∠7 and ∠5 ∠6 and ∠7 2 1 3 4 6 5 7 8 l m n 5y 3y + 5 – THEA MATH REVIEW– 153 Other congruent angle pairs are formed: ■ Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the transver- sal: ∠3 and ∠6; ∠4 and ∠5. ■ Corresponding angles are angles on corresponding sides of the parallel lines, on corresponding sides of the transversal: ∠1 and ∠5; ∠2 and ∠6; ∠3 and ∠7; ∠4 and ∠8. Example In the diagram below, line l is parallel to line m. Determine the value of x. The two angles labeled are corresponding angle pairs, because they are located on top of the parallel lines and on the same side of the transversal (same relative location). This means that they are con- gruent, and we can determine the value of x by solving the equation: 4x + 10 = 8x – 25 10 = 4x – 25 35 = 4x 8.75 = x We can check our answer by replacing the value 8.75 in for x in the expressions 4x + 10 and 8x – 25: 4(8.75) + 10 = 8(8.75) – 25 45 = 45 Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the prob- lem would be solved in the same way. 4x + 10 l m 8x – 25 n – THEA MATH REVIEW– 154 Area, Circumference, and Volume Formulas Here are the basic formulas for finding area, circumference, and volume. They will be discussed in detail in the following sections. Triangles The sum of the measures of the three angles in a triangle always equals 180 degrees. a b c a + b + c = 180° Circle Rectangle Triangle r l w h b A = lw A = 1 _ 2 bh C = 2πr A = πr 2 Cylinder Rectangular Solid h l V = πr 2 h w r h V = lwh C = Circumference A = Area r = Radius l = Length w = Width h = Height v = Volume b = Base – THEA MATH REVIEW– 155 [...]... increasing each month at a pretty steady rate The graph also shows that New Cars increase at a higher rate and that there are many more New Cars sold per month Try to look for scatter plots with different trends—including: ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ increase decrease rapid increase, followed by leveling off slow increase, followed by rapid increase rise to a maximum, followed by a decrease rapid decrease,... two-dimensional object Area is measured in square units, often written as unit2 So, if the length of a triangle is measured in feet, the area of the triangle is measured in feet2 A triangle has three sides, each of which can be considered a base of the triangle A perpendicular line segment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height It measures how tall the triangle... congruent triangles as its bases Height of prism Base of prism Note: This can be confusing The base of the prism is the shape of the polygon that forms it; the base of a triangle is one of its sides 161 – THEA MATH REVIEW – Volume is the amount of space inside a three-dimensional object Volume is measured in cubic units, often written as unit3 So, if the edge of a triangular prism is measured in feet,... that has a polygon as one base, and instead of a matching polygon as the other, there is a point Each of the sides of a pyramid is a triangle Pyramids are also named for the shape of their (non-point) base 1 The volume of a pyramid is determined by the formula ᎏ3ᎏAbh Example Determine the volume of a pyramid whose base has an area of 20 square feet and stands 50 feet tall Since the area of the base is... that has rectangles as bases Recall that the formula for any prism is V = Abh Since the area of the rectangular base is A = lw, we can replace lw for Ab in the formula giving us the more common, easier to remember formula, V = lwh If a prism has a different quadrilateral-shaped base, use the general prisms formula for volume Note: A cube is a special rectangular prism with six congruent squares as sides... represent the legs and c represents the hypotenuse This theorem makes it easy to find the length of any side as long as the measure of two sides is known So, if leg a = 1 and leg b = 2 in the triangle below, it is possible to find the measure of the hypotenuse, c c 1 2 a2 + b2 = c2 12 + 22 = c2 1 + 4 = c2 5 = c2 ͙5 = c ෆ P YTHAGOREAN T RIPLES Sometimes, the measures of all three sides of a right triangle... of one right triangle are congruent to the hypotenuse and leg of another right triangle, the triangles are congruent Hypotenuse-Leg (Hy-Leg) 9' 7' 7' 30˚ 30˚ 7' ≅ ≅ 50˚ 30˚ 5' 5' 30˚ SAS ≅ SAS 7' SSS ≅ SSS 10' ≅ 30˚ 50˚ 30˚ ≅ 10' 50˚ 6' 6' Hy-Leg ≅ Hy-Leg AAS ≅ AAS 159 7' 5' 5' ASA ≅ ASA 7' 9' 50˚ 5' 5' ≅ – THEA MATH REVIEW – Example Determine if these two triangles are congruent 8" 150˚ 150˚ 8" 6" 6"... REVIEW – A prism that has circles as bases is called a cylinder Recall that the formula for any prism is V = Abh Since the area of the circular base is A = πr2 , we can replace πr2 for Ab in the formula, giving us V = πr2h, where r is the radius of the circular base, and h is the height of the cylinder Cylinder r h V = πr 2 h A sphere is a three-dimensional object that has no sides A basketball is a good... b is the base of the triangle, and h is the height Example Determine the area of the triangle below 5" 10" 1 A = ᎏ2ᎏbh 1 A = ᎏ2ᎏ(5)(10) A = 25 in2 V OLUME F ORMULAS A prism is a three-dimensional object that has matching polygons as its top and bottom The matching top and bottom are called the bases of the prism The prism is named for the shape of the prism’s base, so a triangular prism has congruent... LASSIFYING T RIANGLES It is possible to classify triangles into three categories based on the number of congruent (indicated by the symbol: ≅) sides Sides are congruent when they have equal lengths Scalene Triangle Isosceles Triangle Equilateral Triangle no sides congruent more than 2 congruent sides all sides congruent It is also possible to classify triangles into three categories based on the measure . equals 360 degrees. Example m∠1 + m∠2 = 180° m∠1 = m∠3 + m∠5 m∠3 + m∠4 = 180° m∠4 = m∠2 + m∠5 m∠5 + m 6 = 180° m 6 = m∠3 + m∠2 m∠1 + m∠4 + m 6 = 360 ° C LASSIFYING TRIANGLES It is possible to classify. 90-degree angle symbolizes that the triangle is, in fact, a right triangle. Hypotenuse Leg Leg 60 ° 60 ° 60 ° x x x 66 48° 48° – THEA MATH REVIEW– 157 Pythagorean Theorem The Pythagorean theorem. SAS 5' 5' 7' 7' 30˚ 30˚ ASA ≅ ASA 5' 5' SSS ≅ SSS 5' 5' 7' 7' 30˚ 30˚ AAS ≅ AAS 7' 7' Hy-Leg ≅ Hy-Leg 6& apos; 6& apos; 10' 10' 50˚ 50˚ 9' 9' 50˚ 50˚