resources Symbols < is less than > is greater than is less than or equal to is greater than or equal to = is equal to is not equal to

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1 Symbols < is less than > is greater than is less than or equal to is greater than or equal to resources = is equal to is not equal to is approximately equal to similar a absolute value: = ; - = (x, y) perpendicular parallel ordered pair 015 College Board. All rights reserved. AB sur AB uur AB AB A m A arc AB line AB ray AB line segment AB angle A measure of angle A ABC triangle ABC π pi; π.14 ; π = 7 Resources 67

2 Formulas Perimeter P Rectangle Square Circumference = sum of the lengths of the sides P = l + w P = 4s C = πr, C = πd Area Circle Parallelogram Rectangle Square A = π r A = bh A = lw A = s Triangle A = 1 bh Trapezoid A = 1 h( b + b ) 1 Regular Polygon A = 1 ap, where a is the apothem and p is the perimeter Cube Rectangular Prism Cylinder Surface Area SA = 6 e SA = lw + lh + wh SA = π r + πrh 015 College Board. All rights reserved. Cone SA = π r + πrl Regular Pyramid SA = B + 1 pl Sphere SA = 4π r 68 SpringBoard Mathematics Geometry

3 Volume Cylinder Rectangular Prism V = Bh, B = π r V = lwh Triangular Prism V = Bh, B = 1 bh Pyramid V = 1 Bh Cone V = 1 πr h Sphere V = 4 πr Quadratic Equations Standard form ax + bx + c = 0 Quadratic Formula 4 x = b ± b ac a Linear Equations Slope m = y y x x 1 1 Slope-intercept form y = mx + b Point-slope form y - y 1 = m(x x 1 ) 015 College Board. All rights reserved. Standard form Pythagorean Theorem Ax + By = C Other Formulas a + b = c, where c is the hypotenuse of a right triangle 1 1 Distance d = ( x x ) + ( y y ) Midpoint Direct variation x + x y + y m =, y = kx 1 1 Inverse variation y = k x Resources 69

4 Trigonometry For an acute angle A in a right triangle: sin A = cos A = opposite hypotenuse adjacent hypotenuse tan A = opposite adjacent Temperature Celsius C = 5 ( F ) 9 Fahrenheit F = 9 C College Board. All rights reserved. 640 SpringBoard Mathematics Geometry

5 Properties of Real Numbers 015 College Board. All rights reserved. Reflexive Property of Equality For all real numbers a, a = a. Symmetric Property of Equality For all real numbers a and b, if a = b, then b = a. Transitive Property of Equality For all real numbers a, b, and c, if a = b and b = c, then a = c. Substitution Property of Equality For all real numbers a and b, if a = b, then a may be replaced by b. Additive Identity For all real numbers a, a + 0 = 0 + a = a. Multiplicative Identity For all real numbers a, a 1 = 1 a = a. Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property of Multiplication over Addition For all real numbers a and b, a + b = b + a. For all real numbers a and b, a b = b a. For all real numbers a, b, and c, (a + b) + c = a + (b + c). For all real numbers a, b, and c, (a b) c = a (b c). For all real numbers a, b, and c, a(b + c) = a b + a c. Additive Inverse For all real numbers a, there is exactly one real number a such that a + (-a) = 0 and (-a) + a = 0. Multiplicative Inverse For all real numbers a and b where a 0, b 0, there is exactly one number b a such that b a a b = 1 and a b = 1. b a Multiplication Property of Zero For all real numbers a, a 0 = 0 and 0 a = 0. Addition Property of Equality For all real numbers a, b, and c, if a = b, then a + c = b + c. Subtraction Property of Equality For all real numbers a, b, and c, if a = b, then a c = b c. Multiplication Property of Equality For all real numbers a, b, and c, if a = b, then a c = b c. Division Property of Equality For all real numbers a, b, and c, c 0 if a = b, then a = b c c. Zero Product Property of Equality For all real numbers a and b, if a b = 0 then a = 0 or b = 0 or both a and b equal 0. Addition Property of Inequality* For all real numbers a, b, and c, if a > b, then a + c > b + c. Subtraction Property of For all real numbers a, b, and c, if a > b, then a c > b c. Inequality* Multiplication Property of Inequality * Division Property of Inequality* For all real numbers a, b, and c, c > 0, if a > b, then a c > b c. For all real numbers a, b, and c, c < 0, if a > b, then a c < b c. For all real numbers a, b, and c, c > 0 if a > b, then a c > b. c *These properties are also true for <,,. For all real numbers a, b, and c, c < 0 if a > b, then a c < b. c Resources 641

6 Properties of Exponents For any numbers a and b and all integers m and n, a m a n = a m+n (a m ) n = a mn (ab) m = a m b m m a n a = am-n, a 0 m m a a ( b) =, b 0 m b a -n = 1 a, a 0 and 1 n a = n an, a 0 a 0 = 1, a 0 Properties of Radicals In the expression a is the radic and, is the root index. n a, is the radical symbol, and n n a = b, if b n = a b is the nth root of a. a b ± c b = ( a ± c) b, where b 0. ( a b)( c b) a b c d = ac bd, where b 0, d 0. = a b c d, where b 0, c 0, d > 0. Table of Measures Customary Distance/Length 1 yard (yd) = feet (ft) = 6 inches (in.) 1 foot (ft) = 1 inches (in.) 1 mile (mi) = 1760 yards (yd) 1 mile (mi) = 580 feet (ft) 1 acre (ac) = 4,560 square feet (ft) Metric 1 meter (m) = 100 centimeters (cm) 1 meter (m) = 1000 millimeters (mm) 1 meter (m) = 10 decimeters (dm) 1 centimeter (cm) = 10 millimeters (mm) 1 kilometer (km) = 1000 meters (m) Volume 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = cups (c) 1 quart (qt) = 4 cups (c) = pints (pt) 1 gallon (gal) = 16 cups (c) = 4 quarts (qt) Weight/Mass 1 pound (lb)= 16 ounces (oz) 1 ton (T) = 000 pound (lb) 1 liter (L) = 1000 milliliters (ml) 1 liter (L) = 1000 cubic centimeters ( cm ) 1 liter (L) = 100 centiliters (cl) 1 liter (L) = 10 deciliters (dl) 1 kiloliter (kl) = 1000 liters (L) 1 gram (g) = 1000 milligrams (mg) 1 kilogram (kg) = 1000 grams (g) 1 tonne (t) = 1000 kilograms (kg) 015 College Board. All rights reserved. Time 1 minute (min) = 60 seconds (sec) 1 hour (hr) = 60 minutes (min) 1 day (d) = 4 hours (hr) 1 year (yr) = 65 days (d) 1 year (yr) = 5 weeks (wk) 1 year (yr) = 1 months (mo) 64 SpringBoard Mathematics Geometry