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Chapter 5 Perimeter and Area

Chapter 5 Perimeter and Area. 5-1: Perimeter and Area 5-2: Areas of Triangles, Parallelograms, and Trapezoids 5-3: Circumferences and Areas of Circles 5-4:The Pythagorean Theorem 5-5: Special Triangles and Areas of Regular Polygons 5-6: The Distance Formula and the Method of Quadrature

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Chapter 5 Perimeter and Area

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  1. Chapter 5Perimeter and Area 5-1: Perimeter and Area 5-2: Areas of Triangles, Parallelograms, and Trapezoids 5-3: Circumferences and Areas of Circles 5-4:The Pythagorean Theorem 5-5: Special Triangles and Areas of Regular Polygons 5-6: The Distance Formula and the Method of Quadrature 5-7: Proofs Using Coordinate Geometry 5-8: Geometric Probability

  2. 5.1 Perimeter and Area • Perimeter—the distance around an object • To find the perimeter of a polygon, add the lengths of all of its sides.

  3. Area—the surface encompassed by a polygon Area of Triangle=1/2bh, where height, h, is perpendicular to the base

  4. Area of Square=s2 where s is the length of the side

  5. Area of Rectangle = l x w, where l is the length and w is the width

  6. 5.2 Areas of Triangles, Parallelograms, and Trapezoids • Area of Triangle=1/2bh, where h is perpendicular to the base, b; h does not have to touch the base

  7. Area of Parallelogram=b x h, where height, h, is perpendicular to base, b

  8. Area of Trapezoid=1/2(h)(b1+b2) or A=h(midsegment); h is height perpendicular to and intersecting both bases (b1and b2)

  9. 5.3 Circumferences and Areas of Circles • Circumference—perimeter of a circle • C=2πr or C=2d, where r is the radius and d is the diameter

  10. Area of Circle =πr2If they give you the area or circumference and ask for the radius, you must use algebra to solve for r.

  11. 5.4 The Pythagorean Theorem • Pythagorean Theorem: Given a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse • hyp2=leg2+leg2 or c2=a2+b2 • Your longest leg is always c!

  12. Pythagorean Triples: Sets of lengths that always make a right triangle3,4,55,12, 137,24,259, 40, 4111, 60, 61

  13. Using Side Lengths to Determine if a Triangle is Right, Acute, or ObtuseConverse of the Pythagorean Theorem:If c2=a2+b2, then you have a right triangle.If c2<a2+b2, then you have an acute triangle.If c2>a2+b2, then you have an obtuse triangle.

  14. 5.5 Special Triangles and Areas of Regular Polygons • Special Triangle 1: 30-60-90 • Take an equilateral triangle with sides of length 2a and split it in half. This leaves you with a 30-60-90 triangle. • Since everything was split in half, the base of this 30-60-90 triangle is half of the original, or a. • Use the Pythagorean Theorem to get the height:

  15. a√3 Special Triangle 1: 30-60-90 • So, in a 30°-60°-90° triangle, the side opposite 30° is a, the side opposite 60° is a√3, and the side opposite 90° is 2a • When solving problems with a 30-60-90 triangle, set the known side length equal to its ratio. • Solve for a. • Then substitute into the other ratios to find the missing side lengths.

  16. Special Triangle 2: 45-45-90 • Take a 45-45-90 triangle with sides of length a. (Since the angles are congruent, this triangle is isosceles. • Use the Pythagorean Theorem to find the length of the hypotenuse.

  17. Special Triangle 2: 45-45-90 So, in a 45°-45°-90° triangle, the sides opposite 45° are a, and the side opposite 90° is a√2 • When solving problems with a 45-45-90 triangle, set the known side length equal to its ratio. • Solve for a. • Then substitute into the other ratios to find the missing side lengths.

  18. Finding the Area of a Regular Polygon • Let’s say we have a hexagon with side length 10. • Put the hexagon into a circle. How many triangles do we get? What are the angles inside? • Draw the height of the triangles (from the vertex of the circle to the middle of the edge). This is called your apothem in a regular polygon.

  19. Finding the Area of a Regular Polygon • 3. Use special triangles to find the length of the apothem and calculate the area of the triangle. • 4. Multiply this area by the number of triangles in the hexagon to get the total area. Our side length=10, so a=5, and the height must be 5√3.

  20. Finding the Area of a Regular Polygon Formula for the area of a regular polygon: • A=1/2(apothem)(perimeter) • p=(number of sides)(length of side)

  21. 5.6 The Distance Formula and the Method of Quadrature • Distance—the length between two points • Given points (x1, y1) and (x2, y2) Choose point one and point two and substitute in this formula to find distance.

  22. 5.7 Proofs Using Coordinate Geometry • Midpoint—the point which divides a segment into two congruent parts • Given endpoints (x1, y1) and (x2, y2) Slope= • Use properties of polygons to find determine lengths of sides and points of vertices.

  23. 5.8 Geometric Probability • Probability is the likelihood that an event will happen. It is always between 0 and 1. (0 means that it is impossible, and 1 means that it always has to happen.) • P= (#favorable outcomes) (#possible outcomes) • In geometry, probability is usually the percent of area an “event” represents—like the area represented by blue on a spin dial.

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