ACT Math : Triangles

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #89 : Right Triangles

The lengths of the sides of a right triangle are consecutive integers, and the length of the shortest side is . Which of the following expressions could be used to solve for ?

Possible Answers:

Correct answer:

Explanation:

Since the lengths of the sides are consecutive integers and the shortest side is , the three sides are , , and .

We then use the Pythagorean Theorem:

 

 

Example Question #91 : Sat Mathematics

Square  is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point  has coordinates and point  has the coordinates .

Quantity A:  5\sqrt{2}

Quantity B: The distance between points  and

Possible Answers:

The relationship cannot be determined from the information provided.

 

Quantity B is greater.

 

The two quantities are equal.

 

Quantity A is greater.

 

Correct answer:

The two quantities are equal.

 

Explanation:

To find the distance between points  and , split the square into two 45-45-90 triangles and find the hypotenuse. The side ratio of the 45-45-90 triangle is , so if the sides have a length of 5, the hypotenuse must be 5\sqrt{2}.

Example Question #121 : Act Math

What is the diagonal of a computer screen that measures  inches tall by  inches wide?

Possible Answers:

Correct answer:

Explanation:

Plug the values into the Pythagorean Theorem

,

because we are solving for the diagonal we are looking for .

.

Example Question #71 : Right Triangles

A right triangle has legs of length  and , what is the length of the hypotenuse?

Possible Answers:

Correct answer:

Explanation:

To find the hypotenuse of a right triangle, use the Pythagorean Theorem and plug the leg values in for  and :

Example Question #72 : Right Triangles

A right triangle has a base of six and a height of eight. Using this information find the hypotenuse.

Possible Answers:

Correct answer:

Explanation:

This question calls for us to use the Pythagorean Theorem. This theorem has a formula of 

 

where a and b are the sides of a right triangle, adjacent to the right angle, and c is the hypotenuse.

Given our information 

.

Example Question #73 : Right Triangles

A right triangle has a base of seven and a height of twelve. Using this information find the hypotenuse.

Possible Answers:

Correct answer:

Explanation:

This question calls for us to use the Pythagorean Theorem. This theorem has a formula of 

 

where a and b are the sides of a right triangle, adjacent to the right angle, and c is the hypotenuse.

Given our information 

.

Example Question #74 : Right Triangles

Marcus Absent is marking out some lines for a large canvas tent. He paces out  to the north, places a peg, then turns east and paces another  to place another peg. Stopping, he realizes he has forgotten the other pegs, so he makes a beeline for his original starting point.

How many  does Marcus travel to get back to the pegs?

Possible Answers:

Correct answer:

Explanation:

The Pythagorean Theorem states that for any right triangle with legs  and  and hypotenuse :

.

Applying this to Marcus's steps, we know that .

Expand:

So, . To find , just take the square root:

So, Marcus travels exactly  back to the start.

Example Question #93 : Sat Mathematics

Justin travels  to the east and  to the north. How far away from his starting point is he now?

Possible Answers:

Correct answer:

Explanation:

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that  

  

 

Example Question #97 : Sat Mathematics

Susie walks north from her house to a park that is 30 meters away. Once she arrives at the park, she turns and walks west for 80 meters to a bench to feed some pigeons. She then walks north for another 30 meters to a concession stand. If Susie returns home in a straight line from the concession stand, how far will she walk from the concession stand to her house, in meters?

Possible Answers:

70

200

100

25

50

Correct answer:

100

Explanation:

Susie walks 30 meters north, then 80 meters west, then 30 meters north again. Thus, she walks 60 meters north and 80 meters west. These two directions are 90 degrees away from one another.

At this point, construct a right triangle with one leg that measures 60 meters and a second leg that is 80 meters.

You can save time by using the 3:4:5 common triangle. 60 and 80 are  and , respectively, making the hypotenuse equal to .

We can solve for the length of the missing hypotenuse by applying the Pythagorean theorem:

Substitute the following known values into the formula and solve for the missing hypotenuse: side .

 

Susie will walk 100 meters to reach her house.

Example Question #98 : Sat Mathematics

The lengths of the sides of a triangle are consecutive odd numbers and the triangle's perimeter is 57 centimeters. What is the length, in centimeters, of its longest side?

Possible Answers:

17

25

23

19

21

Correct answer:

21

Explanation:

First, define the sides of the triangle. Because the side lengths are consecutive odd numbers, if we define the shortest side will be as , the next side will be defined as , and the longest side will be defined as . We can then find the perimeter of a triangle using the following formula:

Substitute in the known values and variables.

Subtract 6 from both sides of the equation.

Divide both sides of the equation by 3. 

Solve.

This is not the answer; we need to find the length of the longest side, or 

Substitute in the calculated value for  and solve.

The longest side of the triangle is 21 centimeters long.

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