8th Grade Math - Apply the Pythagorean Theorem to Find the Distance Between Two Points in a Coordinate System: CCSS.Math.Content.8.G.B.8

In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?

2√5

10√2

6√2

Explanation

Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100√2=c. 10√2=c

Trig_id

If and , how long is side ?

Not enough information to solve

Explanation

This problem is solved using the Pythagorean theorem $(a^{2}+b^{2}=c^{2})$ . In this formula and are the legs of the right triangle while is the hypotenuse.

Using the labels of our triangle we have:

$o^{2}+a^{2}=h^{2}$

$\dpi{100} h^{2}=(10ft)^{2} +(17ft)^{2}$

$h^{2}=100ft^{2}+289ft^{2}$

$h=\sqrt{389ft^{2}}$

$\rightarrow 19.7 ft$

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

$5.8\textup{ units}$

$6.3\textup{ units}$

$6.7\textup{ units}$

$5.4\textup{ units}$

$5.2\textup{ units}$

Explanation

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

5 2

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:

$\sqrt{34}=\sqrt{c^2}$

To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?

5 miles

25 miles

9 miles

15 miles

7 miles

Explanation

Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.

Parallelogram2

Give the perimeter of the above parallelogram if .

$48\sqrt{3}$

$24+24 \sqrt{3}$

$24+24 \sqrt{2}$

$48\sqrt{2}$

$32+16\sqrt{3}$

Explanation

By the $30^{\circ}-60^{\circ}-90^{\circ}$ Theorem:

$AB = CD= BD\cdot \frac{\sqrt{3}}{3}= 24 \cdot \frac{\sqrt{3}}{3}= 8\sqrt{3}$ , and

$BC = AD = 2\cdot AB = 2 \cdot 8\sqrt{3} = 16 \sqrt{3}$

The perimeter of the parallelogram is

$AB + CD + BC + AD = 8\sqrt{3}+ 8\sqrt{3}+ 16\sqrt{3}+16\sqrt{3} = 48\sqrt{3}$

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

$9.9\textup{ units}$

$8.7\textup{ units}$

$10\textup{ units}$

$11.2\textup{ units}$

$12.3\textup{ units}$

Explanation

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

2 2

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:

$\sqrt{98}=\sqrt{c^2}$

You leave on a road trip driving due North from Savannah, Georgia, at 8am. You drive for 5 hours at 60mph and then head due East for 2 hours at 50mph. After those 7 hours, how far are you Northeast from Savannah as the crow flies (in miles)?

$\sqrt{100000}$

$\sqrt{90000}$

$\sqrt{10000}$

$\sqrt{80000}$

$\sqrt{70000}$

Explanation

Distance = hours * mph

North Distance = 5 hours * 60 mph = 300 miles

East Distance = 2 hours * 50 mph = 100 miles

Use Pythagorean Theorem to determine Northeast Distance

3002 + 1002 =NE2

90000 + 10000 = 100000 = NE2

NE = √100000

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

$12.2\textup{ units}$

$12\textup{ units}$

$11.8\textup{ units}$

$11.5\textup{ units}$

$11.1\textup{ units}$

Explanation

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

7 7

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:

$\sqrt{149}=\sqrt{c^2}$

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

$7.8\textup{ units}$

$7.9\textup{ units}$

$8.2\textup{ units}$

$8.4\textup{ units}$

$8.5\textup{ units}$

Explanation

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

8 2

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs: