Thea practice test_6 pps

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 [...]... them to the next practice test 174 C H A P T E R 6 THEA Writing Review CHAPTER SUMMARY This chapter covers the topics that will help you succeed on the multiple-choice and essay portions of the Writing test You will learn about grammar, organization, as well as how to recognize your audience U nlike math, writing is flexible There are many different ways to convey the same meaning The THEA Writing section... month to month In fact, practice by comparing the total sales of July with October In order to do this, first find out how many cars were sold in each month There were 235 cars sold in July (155 + 80 = 235) and 405 cars sold in October (265 + 140 = 405) With a little bit of quick arithmetic it can quickly be determined that 170 more cars were sold during October (405 – 235 = 170) 171 – THEA MATH REVIEW –... right angle is called the hypotenuse and the other sides are called legs The box in the angle of the 90-degree angle symbolizes that the triangle is, in fact, a right triangle Hypotenuse Leg Leg 157 – THEA MATH REVIEW – Pythagorean Theorem The Pythagorean theorem is an important tool for working with right triangles It states: a2 + b2 = c2, where a and b represent the legs and c represents the hypotenuse... right triangle with sides measuring 6, x, and a hypotenuse 10, what is the value of x? 3, 4, 5 is a Pythagorean triple, and a multiple of that is 6, 8, 10 Therefore, the missing side length is 8 158 – THEA MATH REVIEW – C OMPARING T RIANGLES Triangles are said to be congruent (indicated by the symbol: ≅) when they have exactly the same size and shape Two triangles are congruent if their corresponding... 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" Although the triangles are not aligned the same, there are two congruent corresponding sides, and the angle... height must always be perpendicular to (form a right angle with) the base Note that in an obtuse triangle, the height is outside the triangle, and in a right triangle the height is one of the sides 160 – THEA MATH REVIEW – 10 10 8 12 1 The formula for the area of a triangle is given by A = ᎏ2ᎏbh, where b is the base of the triangle, and h is the height Example Determine the area of the triangle below 5"... 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, the... Since the area of the base is given to us, we only need to replace the appropriate values into the formula 1 V = ᎏ3ᎏAbh 1 V = ᎏ3ᎏ(20)(50) 1 V = 33ᎏ3ᎏ 1 The pyramid has a volume of 33ᎏ3ᎏ cubic feet 162 – THEA MATH REVIEW – Polygons A polygon is a closed figure with three or more sides, for example triangles, rectangles, pentagons, etc A B C F E D Shape Circle 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon... An equiangular polygon has congruent angles Interior Angles To find the sum of the interior angles of any polygon, use this formula: S = 180(x – 2)°, where x = the number of sides of the polygon 163 – THEA MATH REVIEW – Example Find the sum of the interior angles in the polygon below: The polygon is a pentagon that has 5 sides, so substitute 5 for x in the formula: S = (5 – 2) ϫ 180° S = 3 ϫ 180° S... D Z 4 AB VW BC CD WX XY DE YZ EA ZV 3 3 5 5 2 = = = = 6 6 10 10 4 Y These two polygons are similar because their angles are congruent and the ratios of the corresponding sides are in proportion 164 – THEA MATH REVIEW – Quadrilaterals A quadrilateral is a four-sided polygon Since a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360 . measure of the greatest angle: Acute Triangle Right Triangle Obtuse Triangle greatest angle is acute greatest angle is 90° greatest angle is obtuse 1 2 3 56 4 – THEA MATH REVIEW– 1 56 ANGLE-SIDE RELATIONSHIPS Knowing. proportion. 3 2 5 3 6 10 6 10 4 5 A B C D E V X W Y Z ЄA = ЄV = 140° ЄB = ЄW = 60 ° ЄC = ЄX = 140° ЄD = ЄY = 100° ЄE = ЄZ = 100° AB VW 3 6 3 6 5 10 5 10 BC WX CD XY DE YZ EA ZV 2 4 ==== – THEA MATH REVIEW– 164 Quadrilaterals A. 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 is an important