### All Common Core: 8th Grade Math Resources

## Example Questions

### Example Question #1 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

Give the perimeter of the above parallelogram if .

**Possible Answers:**

**Correct answer:**

By the Theorem:

, and

The perimeter of the parallelogram is

### Example Question #51 : Plane Geometry

In a rectangle, the width is 6 feet long and the length is 8 feet long. If a diagonal is drawn through the rectangle, from one corner to the other, how many feet long is that diagonal?

**Possible Answers:**

**Correct answer:**

Given that a rectangle has all right angles, drawing a diagonal will create a right triangle the legs are each 6 feet and 8 feet.

We know that in a 3-4-5 right triangle, when the legs are 3 feet and 4 feet, the hypotenuse will be 5 feet.

Given that the legs of this triangle are twice as long as those in the 3-4-5 triangle, it follows that the hypotense will also be twice as long.

Thus, the diagonal in through the rectangle creates a 6-8-10 triangle. 10 is therefore the length of the diagonal.

### Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

If and , how long is side ?

**Possible Answers:**

Not enough information to solve

**Correct answer:**

This problem is solved using the Pythagorean theorem . In this formula and are the legs of the right triangle while is the hypotenuse.

Using the labels of our triangle we have:

### Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Sam and John both start at the same point. Sam walks 30 feet north while John walks 40 feet west. How far apart are they at their new locations?

**Possible Answers:**

**Correct answer:**

Sam and John have walked at right angles to each other, so the distance between them is the hypotenuse of a triangle. The distance can be found using the Pythagorean Theorem.

### Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?

**Possible Answers:**

1 mile

Cannot be determined

√2 miles

2 + √2 miles

2 – √2 miles

**Correct answer:**

2 – √2 miles

First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(1^{2} + 1^{2}) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.

### Example Question #1 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

Max starts at Point *A* and travels 6 miles north to Point *B* and then 4 miles east to Point *C*. What is the shortest distance from Point *A* to Point *C*?

**Possible Answers:**

4√2 miles

5 miles

10 miles

7 miles

2√13 miles

**Correct answer:**

2√13 miles

This can be solved with the Pythagorean Theorem.

6^{2} + 4^{2} = *c*^{2}

52 = *c*^{2}

*c* = √52 = 2√13

### Example Question #31 : Right Triangles

Angela drives 30 miles north and then 40 miles east. How far is she from where she began?

**Possible Answers:**

35 miles

50 miles

60 miles

45 miles

**Correct answer:**

50 miles

By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a^{2}+b^{2}=c^{2}, 30^{2}+40^{2}=c^{2}, c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.

### Example Question #131 : Triangles

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?

**Possible Answers:**

**Correct answer:**5 miles

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} = 3^{2}+ 4^{2} = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.

### Example Question #11 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A park is designed to fit within the confines of a triangular lot in the middle of a city. The side that borders Elm street is 15 feet long. The side that borders Broad street is 23 feet long. Elm street and Broad street meet at a right angle. The third side of the park borders Popeye street, what is the length of the side of the park that borders Popeye street?

**Possible Answers:**

27.46 feet

22.5 feet

16.05 feet

17.44 feet

18.5 feet

**Correct answer:**

27.46 feet

This question requires the use of Pythagorean Theorem. We are given the length of two sides of a triangle and asked to find the third. We are told that the two sides we are given meet at a right angle, this means that the missing side is the hypotenuse. So we use a^{2 }+ b^{2 }= c^{2}, plugging in the two known lengths for a and b. This yields an answer of 27.46 feet.

### Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Kathy and Jill are travelling from their home to the same destination. Kathy travels due east and then after travelling 6 miles turns and travels 8 miles due north. Jill travels directly from her home to the destination. How miles does Jill travel?

**Possible Answers:**

**Correct answer:**

Kathy's path traces the outline of a right triangle with legs of 6 and 8. By using the Pythagorean Theorem

miles