PYTHAGOREAN THEOREM

In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.

The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.


PYTHAGOREAN THEOREM

In any right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse,

a^{2}+b^{2}=c^{2}


EXAMPLE 10.40

Rene is setting up a holiday light display. He wants to make a 'tree' in the shape of two right triangles, as shown below, and has two 10-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to each stake. How tall should the pole be?

Solution

Step 1. Read the problem. Draw a picture.
Step 2. Identify what we are looking for. We are looking for the height of the pole.
Step 3. Name what we are looking for. The distance from the base of the pole to either stake is the same as the height of the pole. Let x = the height of the pole.
x = the distance from the pole to stake
Each side is a right triangle. We draw a picture of one of them.
Step 4. Translate into an equation. We can use the Pythagorean Theorem to solve for x.
Write the Pythagorean Theorem. a^{2}+b^{2}=c^{2}
Step 5. Solve the equation. Substitute. x^{2}+x^{2}=10^{2}
Simplify. 2 x^{2}=100
Divide by 2 to isolate the variable. \frac{2 x^{2}}{2}=\frac{100}{2}
Simplify. x^{2}=50
Use the Square Root Property. x=\pm \sqrt{50}
Simplify the radical. x=\pm 5 \sqrt{2}
Rewrite to show two solutions. x=5 \sqrt{2}
\not {x=-5 \sqrt{2}}
Approximate this number to the nearest tenth with a calculator. x \approx 7.1
Step 6. Check the answer.
Check on your own in the Pythagorean Theorem.
Step 7. Answer the question. The pole should be about 7.1 feet tall.


TRY 10.79

The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth of a foot.


TRY 10.80

The distance between opposite corners of a rectangular field is four more than the width of the field. The length of the field is twice its width. Find the distance between the opposite corners. Round to the nearest tenth.


EXAMPLE 10.41

Mike wants to put 150 square feet of artificial turf in his front yard. This is the maximum area of artificial turf allowed by his homeowners association. He wants to have a rectangular area of turf with length one foot less than three times the width. Find the length and width. Round to the nearest tenth of a foot.

Solution

Step 1. Read the problem. Draw a picture.
Step 2. Identify what we are looking for. We are looking for the length and width.
Step 3. Name what we are looking for. Let w = the width of the rectangle.
3w-1 = the length of the rectangle
Step 4. Translate into an equation.
We know the area. Write the formula for the area of a rectangle.
Step 5. Solve the equation. Substitute in the values.
Distribute.
This is a quadratic equation, rewrite it in standard form.
Solve the equation using the Quadratic Formula.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Rewrite to show two solutions.
Approximate the answers using a calculator.
We eliminate the negative solution for the width.
Step 6. Check the answer.
Make sure that the answers make sense.
Step 7. Answer the question. The width of the rectangle is approximately 7.2 feet and the length 20.6 feet.


TRY 10.81

The length of a 200 square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden. Round to the nearest tenth of a foot.


TRY 10.82

A rectangular tablecloth has an area of 80 square feet. The width is 5 feet shorter than the length. What are the length and width of the tablecloth? Round to the nearest tenth of a foot.