Analytic Geometry: Name ______
Pythagorean Theorem and Distance Period _____ Date ______
MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle. (Focus on quadrilaterals, right triangles)
Two friends, Kelsey and Christy, live in a city in which the streets are laid out in a grid system. Kelsey lives on Elizabeth Street and Christy lives on Mary Street as shown. The two friends often meet at the local Starbucks for a cup of java. Each grid square represents one city block.
1. How can you find the vertical distance from Kelsey’s to Starbucks?
2. How far in blocks is it for Kelsey to walk to the coffee shop?
3. How can you find the horizontal distance from Christy’s to Starbucks?
4. How far in blocks is it for Christy to walk to the coffee shop?
5. Kelsey wants to meet Christy at her house so that they can go to the mall together. Kelsey can either walk from her house to Starbucks and then to Christy’s house, or she can walk directly to Christy’s house. Which distance is shorter? Explain you answer.
Applying the Pythagorean Theorem
Right Triangle: A triangle with two sides that are perpendicular and has one ______angle.
Hypotenuse: The ______side that is always across from the ______angle in a right triangle.
Leg of a Right Triangle: One of the two sides of the right triangle that form the ______angle.
Pythagorean Theorem: The sum of the squares of the ______is equal to the square of the length of the ______.
A house painter places the bottom of his 20 foot ladder 12 feet from
a house. The top of the ladder rests against the house.
1. How far up the house does the ladder reach?
2. Suppose the same 20 foot ladder is placed so that it reaches 17 feet up the side of the house. Approximately how far out from the house was the bottom of the ladder placed? State your answer both exactly and approximately rounded to the nearest tenth.
3. Kevin is standing 2 miles due north of the school.
James is standing 4 miles due west of the school.
What is the exact and approximate distance between Kevin and James?
Finding the Distance Between Two Points Using the Pythagorean Theorem
Practice: Find the distance between (5,-7) and (-3, 4)7. Earlier you found the distance between Kevin
and James by using the Pythagorean Theorem, now
find the distance using the distance formula. Kevin
is standing 2 miles due north of the school. James
is standing 4 miles due west of the school.
What is the exact and approximate distance
between Kevin and James?
For questions 8 – 10, use the distance formula, find the distance between the two points.
8. (-2, -1) and (2, 2) 9. (3, -6) and ( 8, 6) 10. (3, 4) and (2, 6)
Homework Name ______
Pythagorean Theorem and Distance Date ______
1. Find the length of side x.
2. If the legs of an isosceles right triangle are 6 units long, find the length of the hypotenuse.
3. Eva Lewis wants to put an underground sprinkler system in her back yard. A drawing of the system is shown below. About how many feet of water pipe will Eva need?
For questions 4 – 6, use the distance formula, find the distance between the two points.
4. (1, 1) and (4, 4) 5. (2, 5) and ( 5, 1) 6. (0, 3) and (2, 6)
7. Farmer Joe lives 3 miles east and 1 mile north of the
town of Smallville at (0,0). Hazeltown is located 1 mile west
and 1 mile south of Smallville and Boo City is located 1 mile east
and 5 miles north of Smallville. What is the shortest distance
between Farmer Joe’s house and the highway between
Hazeltown and Boo City?
For questions 8-10, refer to the map below. Each unit on the map is equivalent to 1,000 feet. Round your answers to the nearest foot. The locations of the points are as follows:
Farmer’s Market (0, 6) Marina (2, -4) Park (6, 7)
Bank (-5, -3) School (9, 3) Post Office (-2, 2)
8. What is the distance from the post office
to the bank?
9. What is the distance from the marina to
the school?
10. What is the distance from the park to
The farmer’s market?
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Answers
1. 2. 3. 242.8 ft total
4. 5. 5 6. 7.
8. 5,831 feet 9. 9,899 feet 10. 7, 6083 feet