The Pythagorean Theorem can be used to find the distance between two points in a real-world situation. We can do this by using a coordinate grid that overlays a diagram of the real-world situation.
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
If a and b are legs and c is the hypotenuse, then
a^{2} + b^{2} = c^{2}
Problem 1 :
Gabriela wants to find the distance between her house on one side of a lake and the beach on the other side. She marks off a third point forming a right triangle, as shown in the figure. The distances in the diagram are measured in meters. Use the Pythagorean Theorem to find the straight-line distance from Gabriela’s house to the beach.
Solution :
Step 1 :
Find the length of the horizontal leg.
The length of the horizontal leg is the absolute value of the difference between the x-coordinates of the points (280, 20) and (10, 20).
|280 - 10| = 270
The length of the horizontal leg is 270 meters.
Step 2 :
Find the length of the vertical leg.
The length of the vertical leg is the absolute value of the difference between the y-coordinates of the points (280, 164) and (280, 20).
|164 - 20| = 144
The length of the vertical leg is 144 meters.
Step 3 :
Let a = 270, b = 144 and c represent the length of the hypotenuse. Use the Pythagorean Theorem to write the relationship between a, b and c.
a^{2} + b^{2} = c^{2}
Step 4 :
Substitute a = 270 and b = 144 and solve for c.
270^{2} + 144^{2} = c^{2}
Simplify.
72,900 + 20,736 = c^{2}
93,636 = c^{2}
Take the square root of both sides.
√93,636 = √c^{2}
306 = c
So, the distance from Jose’ house to the beach is 306 meters.
Problem 2 :
Camp Sunshine is also on the lake. Use the Pythagorean Theorem to find the distance between Gabriela’s house and Camp Sunshine to the nearest tenth of a meter.
Solution :
Step 1 :
Find the length of the horizontal leg.
The length of the horizontal leg is the absolute value of the difference between the x-coordinates of the points (200, 20) and (10, 20).
|200 - 10| = 190
The length of the horizontal leg is 190 meters.
Step 2 :
Find the length of the vertical leg.
The length of the vertical leg is the absolute value of the difference between the y-coordinates of the points (200, 120) and (200, 20).
|120 - 20| = 100
The length of the vertical leg is 100 meters.
Step 3 :
Let a = 190, b = 100 and c represent the length of the hypotenuse. Use the Pythagorean Theorem to write the relationship between a, b and c.
a^{2} + b^{2} = c^{2}
Step 4 :
Substitute a = 190 and b = 100 and solve for c.
190^{2} + 100^{2} = c^{2}
Simplify.
36,100 + 10,000 = c^{2}
46,100 = c^{2}
Take the square root of both sides.
√46,100 = √c^{2}
√46,100 = c
Find the value of √46,100 using calculator and round to the nearest tenth
214.7 ≈ c
So, the distance from Gabriela’a house to the Camp Sunshine is about 214.7 meters.
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