Warm-up 3/17/09 • Find the values of x for which sinx = 0 is true. • Give the amplitude and period of the function y = 3sin4x. • Find sin(sin-1 0.6)
Did You Know? Q. What do bulletproof vests, fire escapes, windshield wipers, and laser printers all have in common?A. All invented by women. F: One roach can live on a piece of gum for 5 years. Q. If you were to spell out numbers, how far would you have to go until you would find the letter "A"?A. One thousand • When a coffee seed is planted, it takes five years to yield it's first consumable fruit. • The common goldfish is the only animal that can see both infra-red and ultra-violet light.
Tennessee is bordered by more states than any other. The eight states are Kentucky, Missouri, Arkansas, Mississippi, Alabama, Georgia, North Carolina and Virginia. • Des Moines has the highest per capita Jello consumption in the U.S • The Western-most point in the contiguous United States is Cape Alava, Washington. • There are only three animals with blue tongues, the Black Bear, the Chow Chow dog and the blue-tongued lizard.
Did you know? • In 1886, Coca-cola was first served at a pharmacy in Atlanta, Georgia for only five cents a glass. A pharmacist named John Pemberton created the formula for Coca-cola • Flamingos are able to fly at a speed of approximately 55 kilometers an hour. In one night they can travel about 600 km • Men are able to read fine print better than women can • On average, 150 couples get married in Las Vegas each day • Spiders usually have eight eyes, but still they cannot see that well • In humans, the epidermal layer of skin, which consists of many layers of skin regenerates every 27 days
Camel's milk does not curdle. • The ball on top of a flagpole is called the truck. • People generally read 25% slower from a computer screen compared to paper • Certain female species of spiders such as the Australian crab spider, sacrifice their bodies as a food source for their offspring • One grape vine produce can produce about 20 to 30 glasses of wine. • The Hubble telescope is so powerful that it is like pointing a beam of light at a dime that is two hundred miles away. • On average people fear spiders more than they do death • Every day, over five billion gallons of water are flushed down toilets in the United States • In one trip, a honey bee visits about 75 flowers • Jupiter is the fastest rotating planet, which can complete one revolution in less than ten hours • A chicken loses its feathers when it becomes stressed
Today’s lesson: Geometry Review
By the end of the lesson today, you should be able to: • Find the distance and midpoint between two points on a coordinate plane. • Prove geometric relationships among points and lines using analytical methods.
10.1: Analytic Geometry Distance Formula vs. Deriving the formula Can use graphing techniques and pythagorean theorem. Mid-point formula: Use the words to make sense of the formula
Ex1 Find the distance between points (-2,8) and (6,2). Answer: 10 units
Ex2 Two children are playing hide and seek on the school playground. One of the children is hiding at the location (-4,3) on a map grid of the playground. The child who is doing the seeking is currently at the location (5, -2). Each side of a square on the playground grid represents 5 feet. How far apart are the two children? Answer: Map distance is 10.3 units; unit = 5 feet, so 51.5 feet.
Ex3 Determine whether quadrilateral ABCD with vertices A(5,3), B(4,-2),C(-1,-2), and D(0,3) is a parallelogram. *Parallelogram* One pair of opposite sides are congruent and parallel. The measures of the two sides are equal, and the slopes are parallel, so ABCD is a parallelogram.
Ex4 Find the coordinates of the midpoint of the segment that has endpoints (4,8) and (5,3). Answer: (1/2, 11/2)
Ex5 Prove that the diagonals of a rectangle are congruent. Use corners: (0,0) (a,0) (0,b) (a,b)
Assignment Summary Quiz Hw: p.620-621 #12-36 Even
Warm-up 3/19/09 • A circle has a radius of 12 inches. Find the degree measure of the central angle subtended by an arc 11.5 inches long. 54.9 ° • Find sin390°. ½ or 0.5 3) Solve z2 – 8z = -14 by completing the square. 4 ± √2 4) If x2 = 16 and y2 = 4, what is the greatest possible value of (x - y)2? 36
Did you know? • Sharks are immune to cancer • Manicuring the nails has been done by people for more than 4,000 years • The study of the iris of the eye is called iridology • Back in 1919, the Russian transplant pioneer Serge Voronoff made headlines by grafting monkey testicles onto human males’. • In 1946, the New York Yankees became the first baseball team to travel by plane • By recycling just one glass bottle, the amount of energy that is being saved is enough to light a 100 watt bulb for four hours
The Mall of America, located in Bloomington, Minnesota is so big that it can hold 24,336 school buses • If you have three quarters, four dimes, and four pennies, you have$1.19. You also have the largest amount of money in coins without being able to make change for a dollar. • In the United States, poisoning is the fourth leading cause of death among children • In 1916, Charlie Chaplin was making $10,000 a week, making him the highest paid actor of his time • It's possible to lead a cow upstairs...but not downstairs. • People that smoke have 10 times as many wrinkles as a person that does not smoke • Thomas Edison was afraid of the dark. (Hence, the light bulb?)
Check, go over hw 10.1 p.620-621 #12-36 Even Questions
By the end of today’s lesson, you should be able to: • Use and determine the standard and general forms of the equation of a circle • Graph circles
From Algebra II…
Definitions (you already know?) • Circle – • Set of all pints in a plane an equal distance from a center point. • Radius – • Distance from the center to any point on the circle.
You read… Learning to read a textbook… Read over p. 623 in your book; notice pictures, vocabulary, etc.
Equation of a circle: If r represents the radius of a circle with its center at the origin, then the equation of the circle can be written in the form: x2 + y2 = r2 The equation of the circle with center that is Not at (0,0) but at another point (h,k) is: (x – h)2 + (y – h)2 = r2 This is the standard form of the equation for a circle.
Application A tree in your garden has a ring of flowers. Each flower is four feet from the center of the tree trunk. You draw a coordinate grid to model your yard. The tree trunk is located at (-4,3). Write an equation that models the ring of flowers. (x - -4) 2 + (y – 3)2 = 42 (x + 4)2 + (y – 3)2 = 16
Try This • Write the equation of a circle whose center is at (5,-3) and whose radius is 5. • Describe the translation that gives you the equation (x + 2)2 + (y – 5)2 = 1.
Using the center and radius to graph a circle Find the center and radius of the circle: (x + 6) 2 + (y – 7) 2 = 25 Find the center and radius of the circle whose equation is (x – 16) 2 + (y + 9) 2 = 144
Describe how you could use a stake and a long piece of rope to mark the perimeter of a circular home that will be 20 feet across.
General Form of Circle Equation: x2 + y2 + Dx + Ey + F = 0 When the equation of a circle is given in general form, it can be rewritten in standard form by completing the square for the terms in x and the terms in y. Example…
Example The equation of a circle is x2 + y2 + 4x – 6y + 4 = 0. a. Write the standard form of the equation. b. Find the radius and the coordinates of the center.
x2 + y2 + 4x – 6y + 4 = 0 Group to form perfect square trinomials. (x2 + 4x + ?) + (y2 – 6y + ?) = -4 (x2 + 4x + 4) + (y2 – 6y + 9) = -4 + 4 + 9 Complete the square. (x + 2)2 + (y – 3)2 = 9 Factor the trinomials. (x + 2)2 + (y – 3)2 = 32 b. The center of the circle is located at (-2, 3) and the radius is 3.
Now, you try: The equation of a circle is 2x2 + 2y2 – 4x + 12y – 18 = 0 Write the standard form of the equation Find the radius and the coordinates of the center.
Finding the equation of a circle that passes through three points. • Substitute in the (x,y) points for x and y in the general form of the equation • Use matrices to determine what D,E,F are. • Write the equation • Use completing the square to simplify.
Ex) Write the standard form of the equation of the circle that passes through the points at (5,3), (-2,2), and (-1, -5) Plug in all three sets of points for x & y: x2 + y2 + Dx + Ey + F = 0 (5)2 + (3) 2 + D(5) + E(3) + F = 0 (-2) 2 + (2) 2 + D(-2) + E(2) + F = 0 (-1) 2 + (-5) 2 + D(-1) + E(-5) + F = 0
SOLVE THE 3X3 MATRIX: 5D + 3E + F = -34 -2D + 2E + F = -8 -D – 5E + F = -26 D=-4, E = 2, F = -20
NOW, JUST PUT THOSE NUMBERS INTO THE GENERAL FORM AND SOLVE FOR THE CIRCLE: x2 + y2 + Dx + Ey + F = 0 x2 + y2 + -4x + 2y + -20 = 0 AFTER COMPLETING THE SQUARE, YOU GET: (X – 2)2 + (Y + 1)2 = 25
Now, you try Write the standard form of the equation of the circle that passes through the points (-2, 3), (6, -5), and (0, 7). Then identify the center and the radius of the circle.
Substitute each ordered pair for (x, y) in x2 + y2 + Dx + Ey + F = 0, to create a system of equations. (-2)2 + (3)2 + D(-2) + E(3) + F = 0 (x, y) = (-2, 3) (6)2 + (-5)2 + D(6) + E(-5) + F = 0 (x, y) = (6, -5) (0)2 + (7)2 + D(0) + E(7) + F = 0 (x, y) = (0, 7) Simplify the system of equations. -2D + 3E + F + 13 = 0 6D – 5E + F + 61 = 0 7E + F + 49 = 0 The solution to the system is D = -10, E = -4, and F = -21.
The equation of the circle is x2 + y2 – 10x – 4y – 21 = 0. After completing the square, the standard form is (x – 5)2 + (y – 2)2 = 50. The center of the circle is (5, 2) and the radius is √50 or 5√2.
Assignment Section 10.2 p. 628-630 #25,28,35-39all, 41a, 43, 48, 55, 58 15 problems total