### All ACT Math Resources

## Example Questions

### Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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?

**Possible Answers:**

11

15

6√2

10√2

2√5

**Correct answer:**

10√2

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

### Example Question #72 : Plane Geometry

Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?

**Possible Answers:**

5√5

6√6

√5

√10

**Correct answer:**

5√5

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

10^{2} + 5^{2} = *x*^{2}

100 + 25 = *x*^{2}

√125 = x, but we still need to factor the square root

√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so

5√5= *x*

### Example Question #51 : Triangles

A square enclosure has a total area of 3,600 square feet. What is the length, in feet, of a diagonal across the field rounded to the nearest whole number?

**Possible Answers:**

75

95

85

60

100

**Correct answer:**

85

In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.

The area of a square is found by multiply the lengths of 2 sides of a square by itself.

So, the square root of 3,600 comes out to 60 ft.

The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.

60^{2} + 60^{2} = C^{2}

the square root of 7,200 is 84.8, which can be rounded to 85

### Example Question #52 : Triangles

If the length of CB is 6 and the angle C measures 45º, what is the length of AC in the given right triangle?

**Possible Answers:**

72

12√2

6√2

9

6

**Correct answer:**

6√2

Pythagorean Theorum

AB^{2} + BC^{2} = AC^{2}

If C is 45º then A is 45º, therefore AB = BC

AB^{2} + BC^{2} = AC^{2}

6^{2} + 6^{2} = AC^{2}

2*6^{2} = AC^{2}

AC = √(2*6^{2}) = 6√2

### Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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)?

**Possible Answers:**

**Correct answer:**

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

300^{2 + }100^{2 }=NE^{2 }

90000 + 10000 = 100000 = NE^{2}

NE = √100000

### Example Question #34 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A square garden has an area of 49 ft^{2}. To the nearest foot, what is the diagonal distance across the garden?

**Possible Answers:**

8

10

7

11

9

**Correct answer:**

10

Since the garden is square, the two sides are equal to the square root of the area, making each side 7 feet. Then, using the Pythagorean Theorem, set up the equation 7^{2 }+ 7^{2 }= the length of the diagonal squared. The length of the diagonal is the square root of 98, which is closest to 10.

### Example Question #82 : Sat Mathematics

A man at the top of a lighthouse is watching birds through a telescope. He spots a pelican 5 miles due north of the lighthouse. The pelican flies due west for 12 miles before resting on a buoy. What is the distance, in miles, from the pelican's current resting spot to the lighthouse?

**Possible Answers:**

**Correct answer:**

We look at the 3 points of interest: the lighthouse, where the pelican started, and where the pelican ended. We can see that if we connect these 3 points with lines, they form a right triangle. (From due north, flying exactly west creates a 90 degree angle.) The three sides of the triangle are 5 miles, 13 miles and an unknown distance. Using the Pythagorean Theorem we get:

### Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

An airplane is 8 miles west and 15 miles south of its destination. Approximately how far is the plane from its destination, in miles?

**Possible Answers:**

**Correct answer:**

A right triangle can be drawn between the airplane and its destination.

Destination

15 miles Airplane

8 miles

We can solve for the hypotenuse, x, of the triangle:

8^{2} + 15^{2} = x^{2}

64 + 225 = x^{2}

289 = x^{2}

x = 17 miles

### Example Question #84 : Sat Mathematics

An 8-foot-tall tree is perpendicular to the ground and casts a 6-foot shadow. What is the distance, to the nearest foot, from the top of the tree to the end of the shadow?

**Possible Answers:**

**Correct answer:**

In order to find the distance from the top of the tree to the end of the shadow, draw a right triangle with the height(tree) labeled as 8 and base(shadow) labeled as 6:

From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:

or you can notice that this is simililar to a 3-4-5 triangle. Since the lengths are just increased by a factor of 2, the hypotenuse that is normally 5 would be 10.

### Example Question #421 : Plane Geometry

In the figure above, is a square and is three times the length of . What is the area of ?

**Possible Answers:**

**Correct answer:**

Assigning the length of *ED* the value of *x*, the value of *AE* will be 3*x*. That makes the entire side *AD* equal to 4*x*. Since the figure is a square, all four sides will be equal to 4*x*. Also, since the figure is a square, then angle *A* of triangle *ABE* is a right angle. That gives triangle *ABE* sides of 3*x*, 4*x* and 10. Using the Pythagorean theorem:

(3*x*)^{2} + (4*x*)^{2} = 10^{2}

9*x*^{2} + 16*x*^{2} = 100

25*x*^{2} = 100

*x*^{2} = 4

*x* = 2

With *x* = 2, each side of the square is 4*x*, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.