Transcript Lesson 3
Rachel Pollan and Elisabeth Waters Area Lesson 1 Area polygonal region – the union of a polygon and its interior polygon polygonal region Lesson 1 Area Postulate 8: The Area Postulate Every polygonal region has a positive number called its area such that (1) congruent triangles have equal areas & (2) the area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts. Squares and Rectangles Lesson 2 Squares and Rectangles Postulate 9: The area of a rectangle is the product of its base and altitude. h altitude (height) b base Corollary to Postulate 9: The area of a square is the square of its side. Triangles Theorems Theorem 38: The are of a right triangle is half the product of its legs 1/2hb Theorem 39: The are of a triangle is half the product of any base and corresponding altitude 1/2hb Corollary to Theorem To Theorem 39: Triangles with equal bases and equal altitudes have equal areas Key Terms Altitude-the height of a triangle or the the distance from a vertex to the line opposite, it’s a perpendicular line segment. Quadrilaterals Theorems The area of a parallelogram is the product of any base and corresponding altitude The area of a trapezoid is half the average of its bases multiplied by its height The Pythagorean Theorem Theorems The Pythagorean Theorem: The square of the hypotenuse in a right triangle is equal to the sum of the squares of its legs Or more commonly known as…. Converse of the Pythagorean Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is a right triangle Heron’s Theorem Semiperimeter – half the perimeter of a triangle S = a+b+c 2 Heron’s Theorem: The area of a triangle with sides of lengths a, b, and c and semiperimeter s is: A = √s(s-a)(s-b)(s-c) Heron’s Theorem Corollary to Heron’s Theorem: The area of an equilateral triangle with sides of length a is: A = a2 √3 4 Radicals To simplify: 1. only add radicals with same radicand (number under radical sign). 2. multiply any radicals 3. a radical expression is simplified if a) all parentheses/terms have been multiplied b) no radicand contains perfect squares c) no term contains a radicand the same as another term Radicals 4. Rules: a) √ab = √a √b b) √a b = √ab c) √a2 = a d) √a + √a = 2√a e) √a √a = a f) a√x b√y = ab√xy Radicals 5. Rationalizing the Denominator a) Definition: to eliminate all radicals from the denominator of a fraction b) Method: multiply “top” and “bottom” of fraction by irrational part of denominator Ex. 3 √2 3√2 √2 √2 2 Radicals Ex. (2 - √3) (4 - √3) (4 + √3) (4 - √3) 11 – 6√3 13 8 - 4√3 – 2√3 + 3 16 – 3 What is the area of each? Which on has a larger area? h=8 h=12 b=13 b=5 ANSWERS 52 square units=area LARGER AREA 30 square units h=8 h=12 b=13 b=5 Find area of each! B1=11in h=4in B2=25in h=30cm b=10cm Find area of each! Answers B1=11in AREA=300 square cm h=4in B2=25in h=30cm AREA=72 square inches b=10cm More Problems How many 6-inch by 6-inch square tiles does it take to cover a rectangular floor 12-feet by 2612-feet? Find the area of a square with diagonal length of √6? Find side length x. x 7 14 6 More Problems: Answers How many 6-inch by 6-inch square tiles does it take to cover a rectangular floor 12-feet by 2612-feet? 1,272 square tiles Find the area of a square with diagonal length of √6? 3 Find side length x. x 7 14 6 49 3 More Problems Determine whether a triangle with sides √67, 3√2, and 7 is a right triangle. Why or why not? What is the area of a triangle with sides 4, 4, and 6? More Problems: Answers Determine whether a triangle with sides √67, 3√2, and 7 is a right triangle. Why or why not? Yes. They correspond with the Pythagorean Theorem. 3√22 + 72 = √672 What is the area of a triangle with sides 4, 4, and 6? 3√7