### All ACT Math Resources

## Example Questions

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

Which of the following sets of sides cannnot belong to a right triangle?

**Possible Answers:**

2, 2√3, 4

6, 7, 8

2, 2, 2√2

5, 12, 13

3, 4 ,5

**Correct answer:**

6, 7, 8

To answer this question without plugging all five answer choices in to the Pythagorean Theorem (which takes too long on the GRE), we can use special triangle formulas. Remember that 45-45-90 triangles have lengths of x, x, x√2. Similarly, 30-60-90 triangles have lengths x, x√3, 2x. We should also recall that 3,4,5 and 5,12,13 are special right triangles. Therefore the set of sides that doesn't fit any of these rules is 6, 7, 8.

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

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:**

2√13 miles

5 miles

10 miles

7 miles

4√2 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 #11 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Which set of side lengths CANNOT correspond to a right triangle?

**Possible Answers:**

3, 4, 5

8, 15, 17

6, 8, 11

7, 24, 25

5, 12, 13

**Correct answer:**

6, 8, 11

Because we are told this is a right triangle, we can use the Pythagorean Theorem, *a*^{2} + *b*^{2} = *c*^{2}. You may also remember some of these as special right triangles that are good to memorize, such as 3, 4, 5.

Here, 6, 8, 11 will not be the sides to a right triangle because 6^{2} + 8^{2} = 10^{2.}

### 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

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

**Possible Answers:**

60 miles

50 miles

35 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.

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.

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:**

16.05 feet

17.44 feet

27.46 feet

18.5 feet

22.5 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.

A right triangle has legs of 15m and 20m. What is the length of the hypotenuse?

**Possible Answers:**

45m

30m

25m

40m

35m

**Correct answer:**

25m

The Pythagorean theorem is a^{2} + b^{2} = c^{2}, where a and b are legs of the right triangle, and c is the hypotenuse.

(15)^{2} + (20)^{2} = c^{2} so c^{2} = 625. Take the square root to get c = 25m

Paul leaves his home and jogs 3 miles due north and 4 miles due west. If Paul could walk a straight line from his current position back to his house, how far, in miles, is Paul from home?

**Possible Answers:**

7

√14

4

25

5

**Correct answer:**

5

By using the Pythagorean Theorem, we can solve for the distance “as the crow flies” from Paul to his home:

3^{2} + 4^{2} = *x*^{2}

9 + 16 = *x*^{2}

25 = *x*^{2}

5 = *x*

Given a right triangle where the two legs have lengths of 3 and 4 respectively, what is the length of the hypotenuse?

**Possible Answers:**

25

3

5

9

4

**Correct answer:**

5

The hypotenuse can be found using Pythagorean Theorem, which is a^{2 }+ b^{2 }= c^{2}, so we plug in a = 3 and b = 4 to get c.

c^{2 }=25, so c = 5

### Example Question #12 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Length *AB* = 4

Length *BC* = 3

If a similar triangle has a hypotenuse length of 25, what are the lengths of its two legs?

**Possible Answers:**

5 and 25

15 and 20

20 and 25

3 and 4

15 and 25

**Correct answer:**

15 and 20

Similar triangles are in proportion.

Use Pythagorean Theorem to solve for *AC*:

Pythagorean Theorem: *AB*^{2} + *BC*^{2} = *AC*^{2}

4^{2} + 3^{2} = *AC*^{2}

16 + 9 = *AC*^{2}

25 = *AC*^{2}

*AC* = 5

If the similar triangle's hypotenuse is 25, then the proportion of the sides is *AC*/25 or 5/25 or 1/5.

Two legs then are 5 times longer than *AB* or *BC*:

5 * (*AB*) = 5 * (4) = 20

5 * (*BC*) = 5 * (3) = 15

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