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ACT Math : Plane Geometry

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Example Questions

Example Question #131 : Geometry

A right triangle has legs of length 20cm and 99 cm , what is the length of the hypotenuse?

Possible Answers:

10,201 cm

119 cm

$\sqrt{119}$

104cm

101 cm

Correct answer:

101 cm

Explanation:

To find the hypotenuse of a right triangle, use the Pythagorean Theorem and plug the leg values in for and :
$\\a^2 + b^2 = c^2 \\ \newline 20^2 + 99^2 = c^2 \\ \newline 10,201 = c^2 \\ \\ \sqrt{10,201}=\sqrt{c^2}\\ \newline 101 = c$

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Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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

Possible Answers:

100

Correct answer:

Explanation:

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

a^2+b^2=c^2

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

Given our information

$\\6^2+8^2=c^2\\ =36+64=100\\c=\sqrt{100}=10$ .

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Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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

Possible Answers:

$\sqrt{19}$

$\sqrt{17}$

193

$\sqrt{193}$

Correct answer:

$\sqrt{193}$

Explanation:

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

a^2+b^2=c^2

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

Given our information

$\\7^2+12^2=c^2\\=49+144=193\\c=\sqrt{193}$ .

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Example Question #132 : Geometry

Marcus Absent is marking out some lines for a large canvas tent. He paces out $15\textup{ feet}$ to the north, places a peg, then turns east and paces another $25\textup{ feet}$ 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 $\textup{ feet}$ does Marcus travel to get back to the pegs?

Possible Answers:

$5\sqrt{34}\textup{ feet}$

$\textup{There is not enough information to answer the question.}$

$3\sqrt{13}\textup{ feet}$

$40\textup{ feet}$

$3\sqrt{22}\textup{ feet}$

Correct answer:

$5\sqrt{34}\textup{ feet}$

Explanation:

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

a^2 + b^2 = c^2 .

Applying this to Marcus's steps, we know that 15^2 + 25^2 = c^2 .

Expand:

225 + 625 = 850

So, c^2 = 850 . To find , just take the square root:

$\sqrt{850} = \sqrt{25 \cdot 34} = 5\sqrt{34}$

So, Marcus travels exactly $5\sqrt{34}\textup{ feet}$ back to the start.

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Example Question #74 : Right Triangles

Justin travels $15\textup{ feet}$ to the east and $20\textup{ feet}$ to the north. How far away from his starting point is he now?

Possible Answers:

$45\textup{ ft}$

$35\textup{ ft}$

$25\textup{ ft}$

$22\textup{ ft}$

$30\textup{ ft}$

Correct answer:

$25\textup{ ft}$

Explanation:

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that a^2+b^2=c^2

15^2 + 20^2 = c^2

225+400=c^2

625=c^2

25=c

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Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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:

200

100

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 $3\cdot 20$ and $4\cdot 20$ , respectively, making the hypotenuse equal to $5\cdot 20=100$ .

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

$a^{2}+b^{2}=c^{2}$

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

$a= 60,\ b= 80,\ c= ?$

$(60)^{2}+(80)^{2}=c^{2}$

$3600+6400=c^{2}$

$10,000=c^{2}$

c=100

Susie will walk 100 meters to reach her house.

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Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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:

Correct answer:

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 x+2 , and the longest side will be defined as x+4 . We can then find the perimeter of a triangle using the following formula:

$\text{Perimeter}=\text{side}+\text{side}+\text{side}$

Substitute in the known values and variables.

$\text{Perimeter}=x+(x+2)+(x+4)$

57=3x+6

Subtract 6 from both sides of the equation.

57-6=3x+6-6

51=3x

Divide both sides of the equation by 3.

$\frac{51}{3}=\frac{3x}{3}$

Solve.

x=17

This is not the answer; we need to find the length of the longest side, or x+4 .

$\text{Longest side}=x+4$

Substitute in the calculated value for and solve.

$\text{Longest side}=17+4$

$\text{Longest side}=21$

The longest side of the triangle is 21 centimeters long.

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Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Each of the following answer choices lists the side lengths of a different triangle. Which of these triangles does not have a right angle?

Possible Answers:

6,7,12

$2, 2, 2\sqrt{2}$

5, 12, 13

9, 12, 15

Correct answer:

6,7,12

Explanation:

6, 7, 12 cannot be the side lengths of a right triangle. 6^2+ 7^2 does not equal 12^2 . Also, special right triangle 3-4-5, 5-12-13, and 45-45-90 rules can eliminate all the other choices.

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Given a right triangle with a leg length of 6 and a hypotenuse length of 10, find the length of the other leg, x.

Act_math_106

Possible Answers:

Correct answer:

Explanation:

Using Pythagorean Theorem, we can solve for the length of leg x:

x² + 6² = 10²

Now we solve for x:

x² + 36 = 100

x² = 100 – 36

x² = 64

x = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

In a right triangle a hypotenuse has a length of 8 and leg has a length of 7. What is the length of the third side to the nearest tenth?

Possible Answers:

1.0

3.6

2.4

3.9

Correct answer: 3.9

Explanation:

Using the pythagorean theorem, 8²=7²+x². Solving for x yields the square root of 15, which is 3.9

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #131 : Geometry

Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #132 : Geometry

Example Question #74 : Right Triangles

Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

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