Pythagorean Theorem - GED Math

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

Set and , and solve for :

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

Subtract 81 from both sides to isolate :

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

,

so

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

A triangle is right if and only if it satisfies the Pythagorean relationship

where is the measure of the longest side and are the other two sidelengths. We test each of the four sets of lengths, remembering to convert feet to inches by multiplying by 12.

7 inches, 2 feet, 30 inches:

2 feet is equal to 24 inches. The relationship to be tested is

- False

10 inches, 1 foot, 14 inches:

1 foot is equal to 12 inches. The relationship to be tested is

- False

15 inches, 3 feet, 40 inches:

3 feet is equal to 36 inches. The relationship to be tested is

- False

2 feet, 32 inches, 40 inches:

2 feet is equal to 24 inches. The relationship to be tested is

- True

The correct choice is 2 feet, 32 inches, 40 inches.

By the Pythagorean Theorem, if are the legs of a right triangle and is its hypotenuse,

Substitute and solve for :

The perimeter of the triangle is

The hypotenuse of the right triangle can be calculated using the Pythagorean theorem:

Add the three sides:

The Pythagorean Theorem can be used to derive the length of the second leg:

inches

Add the three sides to get the perimeter.

inches.

The Pythagorean Theorem can be used to derive the length of the second leg:

inches.

Use the area formula for a triangle, with the legs as the base and height:

square inches.

The area of a triangle is calcuated using the formula

In a right triangle, the bases can be used for base and height, so solve for by substitution:

The legs measure 8 and 34 inches, respectively; use the Pythagorean Theorem to find the length of the hypotenuse:

inches

Each leg of an isosceles right triangle has length that is the length of the hypotenuse divided by . The hypotenuse has length 80, so each leg has length

.

The perimeter is the sum of the three sides:

inches.To the nearest tenth:

inches.

The perimeter of Quadrilateral is the sum of the lengths of , , , and .

The first two lengths can be found by subtracting known lengths:

The last two segments are hypotenuses of right triangles, and their lengths can be calculated using the Pythagorean Theorem:

is the hypotenuse of a triangle with legs ; it measures

is the hypotenuse of a triangle with legs ; it measures

Add the four sidelengths:

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

We can set up a proportion statement by comparing the large triangle to the smaller of the two in which it is divided. The sides compared are the hypotenuse and the longer side:

Solve for :

The area of the square is 2,500; its sidelength is the square root of this:

This means that the right triangle in the diagram has legs and 40 and hypotenuse 50, so by the Pythagorean Theorem:

The perimeter of Quadrilateral is the sum of the lengths of .

is a side of the square and has length 2.

is the midpoint of , so is half a side of the square; its length is 1.

The other two sides are hypotenuses of right triangles and can be found using the Pythagorean Theorem.

and are the midpoints of and , respectively,both of which are segments with length 2; therefore,

.

is therefore the hypotenuse of a triangle with legs 1 and 1, and its length is

.

Similarly, is the hypotenuse of a triangle with legs 1 and 2, and its length is

.

Add these:

This rounds to 6.7.

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.

Substitute the dimensions.

The answer is .

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:

or

Subtracting 484 from each side of the equation, you get:

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

Rounding, this is , so the triangle's other leg measures .

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:

or

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 .

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.

Write the formula for the area of a triangle.

Substitute the sides.

The answer is:

Write the formula for the Pythagorean Theorem.

Substitute the values into the equation.

Subtract 25 from both sides.

Square root both sides.

The answer is:

Write the Pythagorean Theorem to find the hypotenuse.

Substitute the dimensions.

Square root both sides.

The answer is:

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.