■ In equilateral triangles, all sides are equal and all angles are equal. ■ In a right triangle, the side opposite the right angle is called the hypotenuse. Pythagorean Theorem The Pythagorean theorem is an important tool for working with right triangles. It states: a 2 + b 2 = c 2 ,where a and b represent the legs and c represents the hypotenuse. This theorem allows you to find the length of any side as long as you know the measure of the other two. a 2 + b 2 = c 2 1 2 + 2 2 = c 2 1 + 4 = c 2 5 = c 2 ͙5 ෆ = c Pythagorean Triples In a Pythagorean triple, the square of the largest num- ber equals the sum of the squares of the other two numbers. Example: As demonstrated above: 1 2 + 2 2 = (͙5 ෆ ) 2 1, 2, and ͙5 ෆ are also a Pythagorean triple because: 1 2 + 2 2 = 1 + 4 = 5 and (͙5 ෆ ) 2 = 5. Pythagorean triples are useful for helping you identify right triangles. Some common Pythagorean triples are: 3:4:5 8:15:17 and 5:12:13 Multiples of Pythagorean Triples Any multiple of a Pythagorean triple is also a Pythagorean triple. Therefore, if given 3:4:5, then 9:12:15 is also a Pythagorean triple. Example: If given a right triangle with sides measuring 6, x, and 10, what is the value of x? Because it is a right triangle, use the Pythagorean theorem. Therefore, 10 2 – 6 2 = x 2 100 – 36 = x 2 64 = x 2 8= x 2 1 √ ¯¯¯ 5 Hypotenuse Right Equilateral 60º 60º60º 55 5 –THE SAT MATH SECTION– 132 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 132 45-45-90 Right Triangles A right triangle with two angles each measuring 45° is called an isosceles right triangle. In an isosceles right triangle: ■ The length of the hypotenuse is ͙2 ෆ multiplied by the length of one of the legs of the triangle. ■ The length of each leg is multiplied by the length of the hypotenuse. x = y = × ᎏ 1 1 0 ᎏ = = 5͙2 ෆ 30-60-90 Triangles In a right triangle with the other angles measuring 30° and 60°: ■ The leg opposite the 30-degree angle is half the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg opposite the 30-degree angle.) ■ The leg opposite the 60 degree angle is ͙3 ෆ times the length of the other leg. Example: x = 2 × 7 = 14 and y = 7͙3 ෆ 60° 30° x y 7 60° 30° 2s s s √ ¯¯¯ 3 10͙2 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 10 x y ͙2 ෆ ᎏ 2 45° 45° –THE SAT MATH SECTION– 133 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 133 Triangle Trigonometry There are special ratios we can use with right triangles. They are based on the trigonometric functions called sine, cosine, and tangent. The popular mnemonic to use is: SOH CAH TOA For an angle, θ, within a right triangle, we can use these formulas: sin θ = cos θ = tan θ = TRIG VALUES OF SOME COMMON ANGLES sin cos tan 30° ᎏ 1 2 ᎏ 45° 1 60° ᎏ 1 2 ᎏ ͙3 ෆ Whereas it is possible to solve some right triangle questions using the knowledge of 30-60-90 and 45-45- 90 triangles, an alternative method is to use trigonometry. For example, solve for x below. Using the knowledge that cos 60° = ᎏ 1 2 ᎏ , just sub- stitute into the equation: ᎏ 5 x ᎏ = ᎏ 1 2 ᎏ , so x = 10. Circles A circle is a closed figure in which each point of the cir- cle is the same distance from a fixed point called the center of the circle. Angles and Arcs of a Circle ■ An arc is a curved section of a circle. A minor arc is smaller than a semicircle and a major arc is larger than a semicircle. ■ A central angle of a circle is an angle that has its vertex at the center and that has sides that are radii. ■ Central angles have the same degree measure as the arc it forms. Length of an Arc To find the length of an arc, multiply the circumference of the circle, 2πr,where r = the radius of the circle by the fraction ᎏ 36 x 0 ᎏ , with x being the degree measure of the arc or central angle of the arc. Example: Find the length of the arc if x = 36 and r = 70. L = ᎏ 3 3 6 6 0 ᎏ × 2(π)70 L = ᎏ 1 1 0 ᎏ × 140π L = 14π r x r o M i n o r A r c M a j o r A r c Central Angle 60 o 5 x ͙3 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 ͙3 ෆ ᎏ 3 ͙3 ෆ ᎏ 2 opposite hypotenuse adjacent hypotenuse opposite adjacent To find sin To find cos To find tan Opposite ᎏ Adjacent Adjacent ᎏᎏ Hypotenuse Opposite ᎏᎏ Hypotenuse –THE SAT MATH SECTION– 134 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 134 Area of a Sector The area of a sector is found in a similar way. To find the area of a sector, simply multiply the area of a circle (π)r 2 by the fraction ᎏ 36 x 0 ᎏ , again using x as the degree measure of the central angle. Example: Given x = 60 and r = 8, find the area of the sector. A = ᎏ 3 6 6 0 0 ᎏ × (π)8 2 A = ᎏ 1 6 ᎏ × 64(π) A = ᎏ 6 6 4 ᎏ (π) A = ᎏ 3 3 2 ᎏ (π) Polygons and Parallelograms A polygon is a figure with three or more sides. Terms Related to Polygons ■ Ve r t i c e s are corner points, also called endpoints, of a polygon. The vertices in the above polygon are: A, B, C, D, E, and F. ■ A diagonal of a polygon is a line segment between two nonadjacent vertices. The two diagonals in the polygon above are line segments BF and AE. ■ A regular (or equilateral) polygon has sides that are all equal. ■ An equiangular polygon has angles that are all equal. Angles of a Quadrilateral A quadrilateral is a four-sided polygon. Since a quadri- lateral can be divided by a diagonal into two triangles, the sum of its angles will equal 180 + 180 = 360°. a + b + c + d = 360° Interior Angles To find the sum of the interior angles of any polygon, use this formula: S = 180(x – 2), with x being the number of polygon sides. Example: Find the sum of the angles in the polygon below: S = (5 – 2) × 180 S = 3 × 180 S = 540 Exterior Angles Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equal 360°. b c d e a b d a c FE D B A r x r o –THE SAT MATH SECTION– 135 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 135 Similar Polygons If two polygons are similar, their corresponding angles are equal and the ratio of the corresponding sides are in proportion. Example: These two polygons are similar because their angles are equal and the ratio of the corresponding sides are in proportion. Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. In the figure above, A ෆ B ෆ || C ෆ D ෆ and B ෆ C ෆ || A ෆ D ෆ . A parallelogram has . . . ■ opposite sides that are equal (A ෆ B ෆ = C ෆ D ෆ and B ෆ C ෆ = A ෆ D ෆ ) ■ opposite angles that are equal (m∠a = m∠c and m∠b = m∠d) ■ and consecutive angles that are supplementary (∠a + ∠b = 180°, ∠b + ∠c = 180°, ∠c + ∠d = 180°, ∠d + ∠a = 180°) Special Types of Parallelograms ■ A rectangle is a parallelogram that has four right angles. ■ A rhombus is a parallelogram that has four equal sides. ■ A square is a parallelogram in which all angles are equal to 90° and all sides are equal to each other. Diagonals In all parallelograms, diagonals cut each other into two equal halves. ■ In a rectangle, diagonals are the same length. ■ In a rhombus, diagonals intersect to form 90-degree angles. BC A D BD AC DC A B AC = DB D CB A AB = BC = CD = DA ∠A = ∠B = ∠C = ∠D D C B A AB = BC = CD = DA D A B C AB = CD D A B C 60° 10 4 6 18 120° 60° 120° 5 2 3 9 –THE SAT MATH SECTION– 136 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 136 ■ In a square, diagonals have both the same length and intersect at 90-degree angles. Solid Figures, Perimeter, and Area The SAT will give you several geometrical formulas. These formulas will be listed and explained in this sec- tion. It is important that you be able to recognize the figures by their names and to understand when to use which formulas. Don’t worry. You do not have to mem- orize these formulas. You will find them at the begin- ning of each math section on the SAT. To begin, it is necessary to explain five kinds of measurement: 1. Perimeter. The perimeter of an object is simply the sum of all of its sides. 2. Area. Area is the space inside of the lines defin- ing the shape. 3. Volume. Volume is a measurement of a three- dimensional object such as a cube or a rectangu- lar solid. An easy way to envision volume is to think about filling an object with water. The vol- ume measures how much water can fit inside. 4. Surface Area. The surface area of an object meas- ures the area of each of its faces. The total surface area of a rectangular solid is the double the sum of the areas of the three faces. For a cube, simply multiply the surface area of one of its sides by 6. 5. Circumference. Circumference is the measure of the distance around a circle. Circumference 4 4 Surface area of front side = 16 Therefore, the surface area of the cube = 16 ؋ 6 = 96. = Area 6 7 4 10 Perimeter = 6 + 7 + 4 + 10 = 27 B C A D AC = DB and AC DB –THE SAT MATH SECTION– 137 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 137 . triangle with the other angles measuring 30° and 60°: ■ The leg opposite the 30-degree angle is half the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg opposite. circle, 2πr,where r = the radius of the circle by the fraction ᎏ 36 x 0 ᎏ , with x being the degree measure of the arc or central angle of the arc. Example: Find the length of the arc if x = 36. ͙2 ෆ multiplied by the length of one of the legs of the triangle. ■ The length of each leg is multiplied by the length of the hypotenuse. x = y = × ᎏ 1 1 0 ᎏ = = 5͙2 ෆ 30-60 -90 Triangles In a