1. Find the area of a circle with radius 5.2 cm. ANSWER 84.9 cm2 2. Find the area of a right triangle with hypotenuse 18 in. and side length 8 in. ANSWER 64.5 in.2 3. A rectangle with sides 4 inches and 5 inches is cut out of a square with sides 8 inches. Find the area that remains. ANSWER 44in.2

Vocabulary probability of an event – is a measure of the likelihood the event will occur; 0 ≤ P(event) ≤ 1 geometric probability – is a ratio that involves a geometric measure such as length or area probability of an event – total possible

Vocabulary Geometric Probability probability and length – probability and area –

EXAMPLE 1 Use lengths to find a geometric probability Find the probability that a point chosen at random on PQ is on RS . SOLUTION Length of RS Length of PQ P(Point is on RS) = 4 ( 2) 5 ( 5) – = = 6 10 , = 3 5 0.6, or 60%.

EXAMPLE 2 Use a segment to model a real-world probability MONORAIL A monorail runs every 12 minutes. The ride from the station near your home to the station near your work takes 9 minutes. One morning, you arrive at the station near your home at 8:46. You want to get to the station near your work by 8:58. What is the probability you will get there by 8:58? SOLUTION STEP 1 Find: the longest you can wait for the monorail and still get to the station near your work by 8:58. The ride takes 9 minutes, so you need to catch the monorail no later than 9 minutes before 8:58, or by 8:49. The longest you can wait is 3 minutes (8:49 – 8:46 = 3 min).

Use a segment to model a real-world probability EXAMPLE 2 Use a segment to model a real-world probability STEP 2 Model the situation. The monorail runs every 12 minutes, so it will arrive in 12 minutes or less. You need it to arrive within 3 minutes. The monorail needs to arrive within the first 3 minutes. STEP 3 Find: the probability. P(you get to the station by 8:58) Favorable waiting time Maximum waiting time = 3 12 = = 1 4 The probability that you arrive by 8:58. is 1 4 or 25%. ANSWER

GUIDED PRACTICE for Examples 1 and 2 Find the probability that a point chosen at random on PQ is on the given segment. Express your answer as a fraction, a decimal, and a percent. RT 1. 2. TS 3. PT 4. RQ , 0.1, 10% ANSWER 1 10 1 2 , 0.5, 50% ANSWER 2 5 , 0.4, 40% ANSWER 7 10 , 0.7, 70% ANSWER

EXAMPLE 3 Use areas to find a geometric probability ARCHERY The diameter of the target shown at the right is 80 centimeters. The diameter of the red circle on the target is 16 centimeters. An arrow is shot and hits the target. If the arrow is equally likely to land on any point on the target, what is the probability that it lands in the red circle?

EXAMPLE 3 Use areas to find a geometric probability SOLUTION Find the ratio of the area of the red circle to the area of the target. P(arrow lands in red region) = Area of red circle Area of target = (82) (402) = 64 1600 = 1 25 The probability that the arrow lands in the red region is 1 25 , or 4%. ANSWER

EXAMPLE 4 Estimate area on a grid to find a probability SCALE DRAWING Your dog dropped a ball in a park. A scale drawing of the park is shown. If the ball is equally likely to be anywhere in the park, estimate the probability that it is in the field. SOLUTION STEP 1 Find the area of the field. The shape is a rectangle, so the area is bh = 10 3 = 30 square units.

EXAMPLE 4 Estimate area on a grid to find a probability STEP 2 Find the total area of the park. Count the squares that are fully covered. There are 30 squares in the field and 22 in the woods. So, there are 52 full squares. Make groups of partially covered squares so the combined area of each group is about 1 square unit. The total area of the partial squares is about 6 or 7 square units. So, use 52 + 6.5 = 58.5 square units for the total area.

EXAMPLE 4 Estimate area on a grid to find a probability STEP 3 Write a ratio of the areas to find the probability. P(ball in field) = Area of field Total area of park 30 58.5 = 300 585 = 20 39 The probability that the ball is in the field is about 20 39 , or 51.3%. ANSWER

GUIDED PRACTICE for Examples 3 and 4 In the target in Example 3, each ring is 8 centimeters wide. Find the probability that an arrow lands in a black region. 14 25 = 56% ANSWER In Example 4, estimate the probability that the ball is in the woods. or about 48.7%. 19 39 ANSWER

Daily Homework Quiz 1. Find the probability that a point chosen at random on JN is on KN. ANSWER or 71.4% 5 7 Find the probability that a randomly chosen point in the figure will lie in the shaded region. ANSWER 36.3%

Daily Homework Quiz 3. A bird is equally likely to be anywhere in the garden shown. What is the probability that the bird will be in the vegetables? ANSWER about or 47% 24 51