GED Math - Pythagorean Theorem

What is the area of a right triangle if the hypotenuse is 10, and one of the side lengths is 6?

Explanation

To determine the other side length, we will need to use the Pythagorean Theorem.

Substitute the hypotenuse and the known side length as either or .

Subtract 36 from both sides and reduce.

Square root both sides and reduce.

The length and width of the triangle are now known.

Write the formula for the area of a triangle.

$A=\frac{1}{2}bh$

Substitute the dimensions.

$A=\frac{1}{2}(8)(6) = 24$

The answer is .

You want to build a garden in the shape of a right triangle. If the two arms will be 6ft and 8ft, what does the length of the hypotenuse need to be?

Explanation

You want to build a garden in the shape of a right triangle. If the two arms will be 6ft and 8ft, what does the length of the hypotenuse need to be?

To find the length of a hypotenuse of a right triangle, simply use the Pythagorean Theorem.

Where a and b are the arm lengths, and c is the hypotenuse.

Plug in our knowns and solve.

$c=\sqrt{100ft^2}=10ft$

Note that we could also have found c by identifying a Pythagorean Triple:

3x-4x-5x

3(2)-4(2)-5(2)

6-8-10

The hypotenuse of a right triangle is $1285;\text{in}$ and one of its leg measures $1028;\text{in}$ . What is the length of the triangle's other leg? Round to the nearest hundredth.

$771;\text{in}$

$754.25;\text{in}$

$852.36;\text{in}$

$796.21;\text{in}$

$225.54;\text{in}$

Explanation

For this problem, you just need to remember your handy Pythagorean theorem. Remember that it is defined as:

where and are the legs of the triangle, and is the hypotenuse. Remember, however, that this only works for right triangles. Thus, based on your data, you know:

Subtracting 1056784 from each side of the equation, you get:

Using your calculator to calculate the square root, you get:

The length of the missing side of the triangle is $771;\text{in}$ .

If the hypotenuse of a right triangle is 5, and a side length is 2, what is the area?

$\sqrt{21}$

$3\sqrt7$

$\frac{\sqrt{29}}{2}$

$\frac{\sqrt{21}}{2}$

Explanation

To find the other side length, we will need to first use the Pythagorean Theorem.

Substitute the side and hypotenuse.

Solve for the missing side.

$a=\sqrt{21}$

Write the formula for the area of a triangle.

$A=\frac{BH}{2}$

Substitute the sides.

$A=\frac{2\sqrt{21}}{2} = \sqrt{21}$

The answer is: $\sqrt{21}$

A right triangle has hypotenuse with length 20 and a leg of length 9. The length of the other leg is:

Between 17 and 18.

Between 18 and 19.

Between 16 and 17.

Between 15 and 16.

Explanation

By the Pythagorean Theorem, if we let be the length of the hypotenuse, or longest side, of a right triangle, and and be the lengths of the legs, the relation is

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

Set and , and solve for $b^{2}$ :

$9^{2}+ b^{2} = 20^{2}$

Square the numbers - that is, multiply them by themselves:

$81+ b^{2} =400$

Subtract 81 from both sides to isolate $b^{2}$ :

$81+ b^{2} - 81 =400-81$

$b^{2} = 319$

$b=\sqrt{319}$

To find out what integers falls between, it is necessary to find the perfect square integers that flank 319. We can see by trial and error that

$17^{2}= 17 \times 17 = 289$

$18^{2}= 18 \times 18 = 324$

$17<\sqrt{319}< 18$

The length of the second leg thus falls between 17 and 18.

If the hypotenuse of a right triangle if 7, and a side length is 5, what must be the length of the missing side?

$2\sqrt{6}$

$\frac{7}{5}$

$\frac{5}{7}$

$\sqrt{74}$

$6\sqrt2$

Explanation

Write the formula for the Pythagorean Theorem.

Substitute the values into the equation.

Subtract 25 from both sides.

Square root both sides.

$a = \sqrt{24} = \sqrt{4}\cdot \sqrt{6} = 2\sqrt{6}$

The answer is: $2\sqrt{6}$

Determine the hypotenuse of a right triangle if the side legs are respectively.

$2\sqrt5$

$4\sqrt5$

$8\sqrt2$

Explanation

Write the Pythagorean Theorem to find the hypotenuse.

Substitute the dimensions.

Square root both sides.

$\sqrt{20}=\sqrt{c^2}$

$c= \sqrt{20} = \sqrt{4} \times \sqrt{5}$

The answer is: $2\sqrt5$

A car left City A and drove straight east for miles then it drove straight north for miles, where it stopped. In miles, what is the shortest distance between the car and City A?

Explanation

Start by drawing out what the car did.

You'll notice that a right triangle will be created as shown by the figure above. Thus, the shortest distance between the car and City A is also the hypotenuse of the triangle. Use the Pythagorean Theorem to find the distance between the car and City A.

$\text{Distance}=\sqrt{60^2+80^2}=100$

You are visiting a friend who has right-triangular shaped pool. You are seeing who can swim around the perimeter of the pool fastest. If the long side is 20 meters, and second shortest side is 15 meters long, how long is the shortest side?

$5\sqrt{7}m$