SAT Math Essentials for English_6 potx

There are two rules related to tangents: 1. A radius whose endpoint is on the tangent is always perpendicular to the tangent line. 2. Any point outside a circle can extend exactly two tangent lines to the circle. The distances from the origin of the tangents to the points where the tangents intersect with the circle are equal. Practice Question What is the length of A ෆ B ෆ in the figure above if B ෆ C ෆ is the radius of the circle and A ෆ B ෆ is tangent to the circle? a. 3 b. 3͙2 ෆ c. 6͙2 ෆ d. 6͙3 ෆ e. 12 A B 6 30° C AB = AC — — B C A –GEOMETRY REVIEW– 125 Answer d. This problem requires knowledge of several rules of geometry. A tangent intersects with the radius of a circle at 90°. Therefore, ΔABC is a right triangle. Because one angle is 90° and another angle is 30°, then the third angle must be 60°. The triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the leg opposite the 60° angle is ͙3 ෆ ϫ the leg opposite the 30° angle. In this figure, the leg opposite the 30° angle is 6, so A ෆ B ෆ , which is the leg opposite the 60° angle, must be 6͙3 ෆ .  Polygons A polygon is a closed figure with three or more sides. Example Terms Related to Polygons ■ A regular (or equilateral) polygon has sides that are all equal; an equiangular polygon has angles that are all equal. The triangle below is a regular and equiangular polygon: ■ Vertices are corner points of a polygon. The vertices in the six-sided polygon below are: A, B, C, D, E, and F. B CF A DE –GEOMETRY REVIEW– 126 ■ A diagonal of a polygon is a line segment between two non-adjacent vertices. The diagonals in the polygon below are line segments A ෆ C ෆ , A ෆ D ෆ , A ෆ E ෆ , B ෆ D ෆ , B ෆ E ෆ , B ෆ F ෆ , C ෆ E ෆ , C ෆ F ෆ , and D ෆ F ෆ . Quadrilaterals A quadrilateral is a four-sided polygon. Any quadrilateral can be divided by a diagonal into two triangles, which means the sum of a quadrilateral’s angles is 180° ϩ 180° ϭ 360°. Sums of Interior and Exterior Angles To find the sum of the interior angles of any polygon, use the following formula: S ϭ 180(x Ϫ 2), with x being the number of sides in the polygon. Example Find the sum of the angles in the six-sided polygon below: S ϭ 180(x Ϫ 2) S ϭ 180(6 Ϫ 2) S ϭ 180(4) S ϭ 720 The sum of the angles in the polygon is 720°. 12 4 3 m∠1 + m∠2 + m∠3 + m∠4 = 360° B CF A D E –GEOMETRY REVIEW– 127 Practice Question What is the sum of the interior angles in the figure above? a. 360° b. 540° c. 900° d. 1,080° e. 1,260° Answer d. To find the sum of the interior angles of a polygon, use the formula S ϭ 180(x Ϫ 2), with x being the number of sides in the polygon. The polygon above has eight sides, therefore x ϭ 8. S ϭ 180(x Ϫ 2) ϭ 180(8 Ϫ 2) ϭ 180(6) ϭ 1,080° Exterior Angles The sum of the exterior angles of any polygon (triangles, quadrilaterals, pentagons, hexagons, etc.) is 360°. Similar Polygons If two polygons are similar, their corresponding angles are equal, and the ratio of the corresponding sides is in proportion. Example These two polygons are similar because their angles are equal and the ratio of the corresponding sides is in proportion: ᎏ 2 1 0 0 ᎏ ϭ ᎏ 2 1 ᎏ ᎏ 1 9 8 ᎏ ϭ ᎏ 2 1 ᎏᎏ 8 4 ᎏ ϭ ᎏ 2 1 ᎏ ᎏ 3 1 0 5 ᎏ ϭ ᎏ 2 1 ᎏ 18 30 20 135° 75° 135° 75° 60° 9 15 10 60° 8 4 –GEOMETRY REVIEW– 128 Practice Question If the two polygons above are similar, what is the value of d? a. 2 b. 5 c. 7 d. 12 e. 23 Answer a. The two polygons are similar, which means the ratio of the corresponding sides are in proportion. Therefore, if the ratio of one side is 30:5, then the ration of the other side, 12:d, must be the same. Solve for d using proportions: ᎏ 3 5 0 ᎏ ϭ ᎏ 1 d 2 ᎏ Find cross products. 30d ϭ (5)(12) 30d ϭ 60 d ϭ ᎏ 6 3 0 0 ᎏ d ϭ 2 Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. In the figure above, A ෆ B ෆ || D ෆ C ෆ and A ෆ D ෆ || B ෆ C ෆ . Parallelograms have the following attributes: ■ opposite sides that are equal A ෆ D ෆ ϭ B ෆ C ෆ A ෆ B ෆ ϭ D ෆ C ෆ ■ opposite angles that are equal m∠A ϭ m∠C m∠B ϭ m∠D ■ consecutive angles that are supplementary m∠A ϩ m∠B ϭ 180° m∠B ϩ m∠C ϭ 180° m∠C ϩ m∠D ϭ 180° m∠D ϩ m∠A ϭ 180° AB DC 30 12 5 d –GEOMETRY REVIEW– 129 Special Types of Parallelograms ■ A rectangle is a parallelogram with four right angles. ■ A rhombus is a parallelogram with four equal sides. ■ A square is a parallelogram with four equal sides and four right angles. Diagonals ■ A diagonal cuts a parallelogram into two equal halves. B A CD ᭝ABC = ᭝ADC BA CD AB = BC = DC = AD m∠A = m∠B = m∠C = m∠D = 90 BA CD AB = BC = DC = AD A D AD = BC AB = DC –GEOMETRY REVIEW– 130 ■ In all parallelograms, diagonals cut each other into two equal halves. ■ In a rectangle, diagonals are the same length. ■ In a rhombus, diagonals intersect at right angles. ■ In a square, diagonals are the same length and intersect at right angles. B A CD AC = DB AC DB BA CD AC DB BA CD AC = DB BA E C D AE = CE DE = BE –GEOMETRY REVIEW– 131 Practice Question Which of the following must be true about the square above? I. a ϭ b II. A ෆ C ෆ ϭ B ෆ D ෆ III. b ϭ c a. I only b. II only c. I and II only d. II and III only e. I, II, and III Answer e. A ෆ C ෆ and B ෆ D ෆ are diagonals. Diagonals cut parallelograms into two equal halves. Therefore, the diagonals divide the square into two 45-45-90 right triangles. Therefore, a, b, and c each equal 45°. Now we can evaluate the three statements: I: a ϭ b is TRUE because a ϭ 45 and b ϭ 45. II: A ෆ C ෆ ϭ B ෆ D ෆ is TRUE because diagonals are equal in a square. III: b ϭ c is TRUE because b ϭ 45 and c ϭ 45. Therefore I, II, and III are ALL TRUE.  Solid Figures, Perimeter, and Area There are five kinds of measurement that you must understand for the SAT: 1. The perimeter of an object is the sum of all of its sides. Perimeter ϭ 5 ϩ 13 ϩ 5 ϩ 13 ϭ 36 5 13 13 5 D A a c b C B –GEOMETRY REVIEW– 132 2. Area is the number of square units that can fit inside a shape. Square units can be square inches (in 2 ), square feet (ft 2 ), square meters (m 2 ), etc. The area of the rectangle above is 21 square units. 21 square units fit inside the rectangle. 3. Volume is the number of cubic units that fit inside solid. Cubic units can be cubic inches (in 3 ), cubic feet (ft 2 ), cubic meters (m 3 ), etc. The volume of the solid above is 36 cubic units. 36 cubic units fit inside the solid. 4. The surface area of a solid is the sum of the areas of all its faces. To find the surface area of this solid . . . add the areas of the four rectangles and the two squares that make up the surfaces of the solid. ϭ 1 cubic unit ϭ 1 square unit –GEOMETRY REVIEW– 133 5. Circumference is the distance around a circle. If you uncurled this circle . . . you would have this line segment: The circumference of the circle is the length of this line segment. Formulas The following formulas are provided on the SAT. You therefore do not need to memorize these formulas, but you do need to understand when and how to use them. Circle Rectangle Triangle r l w h b A = lw C = 2πr A = πr 2 Cylinder Rectangle 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 A = 1 2 bh –GEOMETRY REVIEW– 134 [...]... solve for x: y2 Ϫ y1 ᎏᎏ ϭ Ϫ3 x2 Ϫ x 1 0Ϫ3 ᎏᎏ ϭ Ϫ3 xϪ6 Ϫ3 ᎏᎏ ϭ Ϫ3 xϪ6 Ϫ3 (x Ϫ 6) ᎏᎏ ϭ Ϫ3(x Ϫ 6) xϪ6 Simplify Ϫ3 ϭ Ϫ3x ϩ 18 Ϫ3 Ϫ 18 ϭ Ϫ3x ϩ 18 Ϫ18 Ϫ21 ϭ Ϫ3x Ϫ21 Ϫ3x ᎏᎏ ϭ ᎏᎏ Ϫ3 Ϫ3 Ϫ21 ᎏᎏ ϭ x Ϫ3 7ϭx Therefore, the x-coordinate of the y-intercept is 7, so the y-intercept is (7,0) 148 C H A P T E R 8 Problem Solving This chapter reviews key problem-solving skills and concepts that you need to know for the SAT. .. of SAT questions Each sample SAT question is followed by an explanation of the correct answer Translating Words into Numbers To solve word problems, you must be able to translate words into mathematical operations You must analyze the language of the question and determine what the question is asking you to do The following list presents phrases commonly found in word problems along with their mathematical... rectangular solids has dimension closest to that of the circular cylinder? a 3 ϫ 3 ϫ 5 b 3 ϫ 5 ϫ 5 c 2 ϫ 5 ϫ 9 d 3 ϫ 5 ϫ 9 e 5 ϫ 5 ϫ 9 Answer d First determine the approximate volume of the cylinder The formula for the volume of a cylinder is V ϭ πr2h (Because the question requires only an approximation, use π ≈ 3 to simplify your calculation.) V ϭ πr2h V ≈ (3)(32)(5) V ≈ (3)(9)(5) V ≈ (27)(5) V ≈ 135 Now... much greater is the circumference of Ms Stone’s circle than Mr Suarez’s circle? a 3π b 6π c 12π d 108π e 216π Answer c You must determine the circumferences of the two circles and then subtract The formula for the circumference of a circle is C ϭ 2πr Mr Suarez’s circle has a radius of 6: C ϭ 2πr C ϭ 2π(6) C ϭ 12π Ms Stone’s circle has a radius of 12: C ϭ 2πr C ϭ 2π(12) C ϭ 24π Now subtract: 24π Ϫ 12π... the formula: d ϭ ͙(x2 Ϫ xෆy2 Ϫ y1ෆ ෆ1)2 ϩ (ෆ)2 d ϭ ͙(Ϫ3 Ϫෆ(Ϫ4 Ϫෆ2 ෆ 2)2 ϩ ෆ (Ϫ4))ෆ d ϭ ͙(Ϫ3 Ϫෆ(Ϫ4 ϩෆ ෆ 2)2 ϩ ෆ 4)2 d ϭ ͙(Ϫ5)2 ෆ ෆϩ (0)2 d ϭ ͙25 ෆ dϭ5 The distance is 5 Practice Question (Ϫ5,6) (1,Ϫ4) What is the distance between the two points shown in the figure above? a ͙20 ෆ b 6 c 10 d 2͙34 ෆ e 4͙34 ෆ 142 – GEOMETRY REVIEW – Answer d To find the distance between two points, use the following formula:... Ϫ4 Plug the points into the formula: d ϭ ͙(x2 Ϫ xෆy2 Ϫ y1ෆ ෆ1)2 ϩ (ෆ)2 d ϭ ͙(1 Ϫ (Ϫෆෆෆ ෆ5))2 ϩ (Ϫ4 Ϫ 6)2 d ϭ ͙(1 ϩ 5ෆ10)2 ෆ)2 ϩ (Ϫෆ d ϭ ͙(6)2 ϩෆ2 ෆ (Ϫ10)ෆ d ϭ ͙36 ϩ 1ෆ ෆ00 d ϭ ͙136 ෆ d ϭ ͙4 ϫ 34 ෆ d ϭ ͙34 ෆ The distance is 2͙34 ෆ Midpoint A midpoint is the point at the exact middle of a line segment To find the midpoint of a segment on the coordinate plane, use the following formulas: x ϩx 1 2 Midpoint... Ϫy 3 Ϫ (Ϫ2) 3ϩ2 2 1 Slope ϭ ᎏᎏ ϭ ᎏ(ᎏ ϭ ᎏᎏ ϭ ᎏ5ᎏ x2 Ϫ x1 1 Ϫ Ϫ3) 1ϩ3 4 Therefore, the slope of the line is ᎏ5ᎏ 4 Practice Question (5,6) (1,3) 144 – GEOMETRY REVIEW – What is the slope of the line shown in the figure on the previous page? a b c 1 ᎏᎏ 2 3 ᎏᎏ 4 4 ᎏᎏ 3 d 2 e 3 Answer b To find the slope of a line, use the following formula: y2 Ϫ y1 vertical slope ϭ ᎏachangege ϭ ᎏᎏ horizont ᎏ l chan x2 Ϫ x1... unknown point, the x2 x1 y-intercept (x,0), and set up a ratio with the known slope, ᎏ2ᎏ, and solve for x: 3 y2 Ϫ y1 2 ᎏᎏ ϭ ᎏᎏ x2 Ϫ x1 3 0Ϫ4 2 ᎏᎏ ϭ ᎏᎏ xϪ1 3 145 – GEOMETRY REVIEW – 0Ϫ4 2 ᎏᎏ ϭ ᎏᎏ xϪ1 3 Find cross products (Ϫ4)(3) ϭ 2(x Ϫ 1) Ϫ12 ϭ 2x Ϫ 2 Ϫ12 ϩ 2 ϭ 2x Ϫ 2 ϩ 2 10 2x Ϫᎏ2ᎏ ϭ ᎏ2ᎏ 10 Ϫᎏ2ᎏ ϭ x Ϫ5 ϭ x Therefore, the x-coordinate of the y-intercept is Ϫ5, so the y-intercept is (Ϫ5,0) Facts about Slope... a picture to help you better understand: x 10 10 x Based on the figure, you know that the perimeter is 10 ϩ 10 ϩ x ϩ x So set up an equation and solve for x: 10 ϩ 10 ϩ x ϩ x ϭ 42 20 ϩ 2x ϭ 42 20 ϩ 2x Ϫ 20 ϭ 42 Ϫ 20 2x ϭ 22 2x 22 ᎏ ᎏ ϭ ᎏᎏ 2 2 x ϭ 11 Therefore, we know that the length of the other two sides of the rectangle is 11 Practice Question The height of a triangular fence is 3 meters less than... to help you better understand the problem The triangle has a base of 7 meters The height is three meters less than the base (7 Ϫ 3 ϭ 4), so the height is 4 meters: 4 7 135 – GEOMETRY REVIEW – The formula for the area of a triangle is ᎏ1ᎏ(base)(height): 2 A ϭ ᎏ1ᎏbh 2 A ϭ ᎏ1ᎏ(7)(4) 2 A ϭ ᎏ1ᎏ(28) 2 A ϭ 14 The area of the triangular wall is 14 square meters Practice Question A circular cylinder has a radius . 90°. Therefore, ΔABC is a right triangle. Because one angle is 90° and another angle is 30°, then the third angle must be 60 °. The triangle is therefore a 30 -60 -90 triangle. In a 30 -60 -90 triangle,. the circle is the length of this line segment. Formulas The following formulas are provided on the SAT. You therefore do not need to memorize these formulas, but you do need to understand when. 3| B ෆ C ෆ ϭ | 6| B ෆ C ෆ ϭ 6 Practice Question A B C (Ϫ2,7) (Ϫ2, 6) (5, 6) A B C (Ϫ3,3) (Ϫ3,Ϫ2) (3,Ϫ2) –GEOMETRY REVIEW– 140 What is the sum of the length of A ෆ B ෆ and the length of B ෆ C ෆ ? a. 6 b.