Chapter 7 Sect. 2. A pythagorean triple is a set of three nonzero whole numbers a, b, and c, that satisfy the equation a 2 + b 2 = c 2.

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1 Chapter 7 Sect. 2 The well-known right triangle relationship called the Pythagorean Theorem is named for Pythagoras, a Greek mathematician who lived in the sixth century b.c. We now know that the Babylonians, Egyptians, and Chinese were aware of this relationship before its discovery by Pythagoras. A pythagorean triple is a set of three nonzero whole numbers a, b, and c, that satisfy the equation a 2 + b 2 = c 2. If you multiply each number in a Pythagorean triple by the same whole number, the three numbers that result also form a Pythagorean triple. In some cases, you will write the length of a side in simplest radical form. To simplify radicals: 1.) Break the radicand (the number under the radical) into a product that includes a perfect square. 2.) Find the square root of the perfect squares. Leave what is left under the radical. A right triangle has the following measurements. Find the length of the missing side. Do the lengths of the sides form a Pythagorean triple? A.) hypotenuse= 100 leg= 80. B.) leg=350 leg= 250 Find the area of the triangle. You can use the Converse of the Pythagorean Theorem to determine whether a triangle is a right triangle Is this triangle a right triangle? Suppose a triangle has sides of lengths a, b, and c, where c is the length of the longest side. If c 2 > a 2 + b 2, the Converse of the Pythagorean Theorem leads to the conclusion that the angle opposite side c must have measure greater than 90. Thus, the triangle is obtuse. Similarly, if c 2 < a 2 + b 2, the triangle is acute. Example 4 The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right. A. 7, 8, and 9 B. 6, 11, 14 C.12, 13, 15.

2 Chapter 7 Sect. 3 The acute angles of an isosceles right triangle are both 45 angles. Another name for an isosceles right triangle is a triangle. If each leg has length x and the hypotenuse has length y, you can solve for y in terms of x. Find the value of each variable. A. B. A high school softball diamond is a square. The distance from base to base is 60 ft. To the nearest foot, how far does a catcher throw the ball from home plate to second base? Another type of special right triangle is a triangle. Find the value of each variable. A. B. Example 4 A moose warning sign is an equilateral triangle. Each side is 1 m long. Find the area of the sign.to find the area, copy the triangle, draw an altitude h and find its length. The altitude bisects the base and divides the triangle into two triangles.

3 Chapter 7 Sect. 4 You learned that the bases of a trapezoid are the parallel sides and the legs are the nonparallel sides. The height of a trapezoid is the perpendicular distance h between the bases. Approximate the area of Arkansas by finding the area of the trapezoid shown. Find the area of trapezoid PQRS. Leave your answer in simplest radical form. Rhombuses and kites have perpendicular diagonals. This property allows you to find areas using the following theorem Find the area of the following: A.kite B.rhombus

4 Chapter 7 Sect. 5 You can circumscribe a circle about any regular polygon. The center of a regular polygon is the center of the circumscribed circle. The radius is the distance from the center to a vertex. The apothem is the perpendicular distance from the center to a side. The figure at the right is a regular pentagon with radii and an apothem drawn. Find the measure of each numbered angle. Suppose you have a regular n-gon with side s. The radii divide the figure into n congruent isosceles triangles. Each isosceles triangle has area equal to ½ as. Since there are n congruent triangles, the area of the n-gon is A = n ½ as. The perimeter p of the n-gon is ns. Substituting p for ns results in a formula for the area in terms of a and p: A = ap. Find the area of a regular decagon with a 12.3-in. apothem and 8-in. sides. The smaller triangles in a Minneapolis sculpture are equilateral. Each has a 12.7-in. radius. What is the area of each to the nearest square inch?

5 Chapter 7 Sect. 6 Find the measure of each central angle in the circle graph. Identify the following in cirlce O. a. the minor arcs b. the semicircles c. the major arcs that contain point A Find m COD, measure of arc CDA, measure of arc AD, and measure of arc BAD. The circumference of a circle is the distance around the circle. The number pi (π) is the ratio of the circumference of a circle to its diameter. The measure of an arc is in degrees while the arc length is a fraction of a circle's circumference It is possible for two arcs of different circles to have the same measure but different lengths, as shown at the left. It is also possible for two arcs of different circles to have the same length but different measures. Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles. Example 4 Find the length of each arc shown in red. Leave your answer in terms of π. Example 5 A car has a turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. In completing the (outer) turning circle, how much farther does a tire travel than a tire on the concentric inner circle?

6 Chapter 7 Sect. 7 Recall the following for finding area of a circle. A sector of a circle is a region bounded by an arc of the circle and the two radii to the arc's endpoints. You name a sector using one arc endpoint, the center of the circle, and the other arc endpoint. Find the area of sector ZOM. Leave your answer in terms of π A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle. To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed. Find the area of the shaded segment. Round your answer to the nearest tenth.

7 Chapter 7 Sect. 8 You may recall that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes. Sometimes you can use a geometric probability model in which you let points represent outcomes. You find probabilities by comparing measurements of sets of points. For example, if points of segments represent outcomes, then A gnat lands at a random point on the ruler's edge. Find the probability that the point is between 3 and 7. Assume that a dart you throw will land on the 1-ft square dartboard and is equally likely to land at any point on the board. Find the probability of hitting each of the blue, yellow, and red regions. The radii of the concentric circles are 1, 2, and 3 inches, respectively. To win a prize in a carnival game, you must toss a quarter so that it lands entirely within the circle as shown at the left. Find the probability of this happening on one toss. Assume that the center of a tossed quarter is equally likely to land at any point within the 8-in. square.