Geometry Pre-Unit 1 Intro: Area, Perimeter, Pythagorean Theorem, Square Roots, & Quadratics P-1 Square Roots and SRF Square number the product of a number multiplied by itself. 1 * 1 = 1 1 is a square number * = 4 4 is a square number 3 * 3 = 9 9 is a square number List the first twelve square numbers:,,,,,,,,,,, Square Root a number that when multiplied by itself is a square number. 4 is the square root of 16 10 is the square root of 100 18 is the square root of 34 5 is the square root of 65 What are the square roots of the following numbers? A) 5 B) 49 C) 81 D) 11 E) 144 F) 576 G) 900 The following symbol (radical) is used for square roots: Examples: When you see that symbol it means find the square root of the number. 1
Simplest Radical form a radical that is not a perfect square but has been reduced to remove all squares from the radicand. Simplest radical form is a way of breaking up a radicand into primes then square rooting. For each pair of primes in the final breakdown, place one digit in front of the radical as a product and the radicand in the root is multiplied back together in the final result Examples Multiplying Radicals - Multiply what is outside the radical together and multiply what is under the radical together. Examples
Like roots roots with the same radicand P- Adding and Subtracting like roots x and x can be added together to make x, 3y and 3y can be added together to make 6y and in the same way like roots can added together. Examples: 3
P-3 Pythagorean Theorem Pythagorean Theorem- The sides of a right triangle are related according by the equation: a + b = c This is significant because you can find the third side of a right triangle if given the other two sides. a and b are the short sides of the triangle called legs c is the long side called the hypotenuse *c is the side not adjacent to the right angle Example 1: Find the length of the hypotenuse of the right triangle given both legs. Leg = 8in Leg = 6in Example : Find the missing leg of the right triangle given the length of the hypotenuse and a leg. Hypotenuse = 17 Leg = 13in 4
Example 3: Find the length of the hypotenuse given the legs are 6 in. Step 1: Find the hypotenuse using the Pythagorean Theorem 5
P-4 Midpoint and Distance Formulas Midpoint formula: Distance Formula: Find the midpoint of A(-1, 7) and B(4, 13) is calculated using the midpoint formula: The distance between of A(-1, 7) and B(4, 13) is calculated using the distance formula: Class Activity: The following is a mathematical proof Given points: A(, 6) and B(18, 14) Find the midpoint C and prove that it bisects. 6
P-5 Perimeter Perimeter the sum of the lengths of the sides of a polygon (shape) The perimeter of the square at right would be: 1.5in + 1.5in +1.5in + 1.5in = 6in 1.5in The formula for perimeter of a square is P = 4s P stands for perimeter and S stands for side P = 4(1.5) = 6in What are the perimeters of the following squares? A) 3ft on a side B) seven halves inches on a side C).5 yd on a side D) m on a side The formula for perimeter of a rectangle is P = b + h h stands for height and b stands for base b The perimeter of the rectangle is.5 ft h P = (3.ft) + (.5ft) = 6.4 + 5 = 11.4ft 3. ft What would be the perimeter of the following rectangles? Draw a picture first A) base = 3 in B) base = ft C) base = 5.6m height = 8 in height = 3.7 ft height = 3m Extension: The base of the Great Pyramid in Egypt is a square measuring 756ft per side. What is the perimeter of the base of the pyramid? 7
Equilateral Triangle A triangle with all side lengths congruent. What is the perimeter of the figure at right? P = 3(0.5) = 1.5ft 0.5ft What would be the perimeter of an equilateral triangle with: A) side length w B) side length C) side length 14km Isosceles Triangle A triangle with two congruent sides. The triangle at right is isosceles. The congruent sides are called legs. The non-congruent side is called the base. What would the perimeter be of an isosceles triangle with the following measurements? A) legs 4in, base 5in B) legs 1.5ft, base 1ft C) legs ft, base 5.4ft 8
P-6 Circumference & Area Circumference the perimeter of a circle Area the number of square units that it would take to fill an object. Formula s: C = πr or C = dπ A = πr π is about 3.14 Radius half the length of the diameter (r) r = d Diameter the length across a circle through the center (d) A 4ft A 1ft The diameter is 4ft The radius is 1ft C = dπ C = πr A = πr C = 4π 1.56ft C = π 6.8ft A = π 3.14 ft Try these: Find the circumference and area of the circles below given the information: A) radius = 3m B) diameter = 7mi C) radius = in Compound Shapes A shape that contains more than one polygon. The perimeter of this shape is a combination of three sides of a rectangle and a semicircle. Use 3.14 for π Perimeter of the three sides: Perimeter of the semicircle: Sum of the perimeters: 1.3 ft ft 9
P-7 Area of Polygons Area the number of square units that it would take to fill an object. Rectangle: A = bh Square: A = s base = 14km side = 1.5cm A = (3)(14) = 4km height = 3km A = 1.5 =.5cm The area of the rectangle is 4km. The area of the square is 1.5cm. A) Rectangle: b = 4in, h =.5in B) Square s = 1.mi C) Rectangle: b = ft, h = 3.4ft Triangle: A = 0.5 bh h h h b b b Acute triangle Right triangle Obtuse triangle All angles less than 90 One angle exactly 90 One angle larger than 90 Height is inside Height is the edge of the right angle Height is outside the triangle 10
Examples: 3cm 3.1in 1.1ft 4cm 5in.8ft Parallelogram: A = bh b h h b NOTE: The base and height must meet at a perpendicular. 3 in 4ft 4in 5.1ft Trapezoid: A = 0.5h(b1+b) It does not matter which base you label as b1 or b h 11
.5 ft 8ft A = 6 ft 3 ft 34ft A = 48 ft Rhombus: A= 0.5(d1d) d=diagonal Kite A= 0.5(d1d) Solve for the Area of the rhombus if A) JL = 3cm and MK = 6cm B) JZ = 7, LZ = 7, MZ = 5, and KZ = 5 J K Z M L Solve for the Area of the kite if A) WA = 1ft and HT = 34ft B) HS = 4, TS = 16, WS = 5 and AS = 5ft H W S A T 1
If each side of the pentagon at right is congruent and the length of one side is 6m. The apothem is 3.4m. What is the Area of the regular polygon? Regular Polygon: P = Perimeter A = 0.5 ap a = apothem \ a) Find the Area of a regular octagon with the length one side is 10 cm and the apothem is 1.1 cm? b) Find the Area of a regular heptagon with the length one side is 4 cm and the apothem is 3.1 cm? 13
P-8 Using the Quadratic Formula Quadratic Formula If ax bx c a 0, 0, and b 4ac 0, then x b b 4ac Solve by factoring: a Use the quadratic formula to solve: x 6x 8 0 x 7x 15 0 3n 8n 0 3x x 3 0 Solve the quadratic by factoring, by taking the square root, completing the square, or the quadratic formula. n 4n 4 0 14