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SAT Math : Triangles

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Example Questions

Example Question #94 : Geometry

Square 1

The above figure shows Square ABCD . is the midpoint of $\overline{CD}$ ; is the midpoint of $\overline{CN}$ . Construct $\overline{AO}$ .

AB= t . Which of the following expresses the length of $\overline{AO}$ in terms of ?

Possible Answers:

$\frac{5}{4} t$

$\frac{\sqrt{65}}{8} t$

$\frac{\sqrt{17}}{4} t$

$\frac{3}{2}t$

$\frac{\sqrt{5}}{2} t$

Correct answer:

$\frac{5}{4} t$

Explanation:

Since all four sides of a square are congruent,

CD = AD = AB= t

Since is the midpoint of $\overline{CD}$ ,

$CN = ND = \frac{1}{2} CD = \frac{1}{2} t$

Since is the midpoint of $\overline{CN}$ ,

$ON = \frac{1}{2} CN = \frac{1}{2} \cdot \frac{1}{2} t = \frac{1}{4} t$ ,

and

$OD = ON + ND =\frac{1}{4} t + \frac{1}{2} t = \frac{3}{4} t$

$\bigtriangleup ADO$ is a right triangle, so, by the Pythagorean Theorem,

$AO= \sqrt{(AD)^{2}+(OD)^{2} }$

Substituting:

$AO= \sqrt{t^{2}+\left ( \frac{3}{4} t \right )^{2} }$

$AO= \sqrt{t^{2}+ \frac{9}{16} t ^{2} }$

$AO= \sqrt{ \frac{25}{16} t ^{2} }$

Apply the Product of Radicals and Quotient of Radicals Rules:

$AO= \sqrt{ \frac{25}{16} } \cdot\sqrt{ t ^{2}}$

$AO= \sqrt{ \frac{25}{16} } \cdot t$

$AO= \frac{\sqrt{ 25}}{\sqrt{ 16}} \cdot t$

$AO= \frac{5}{4} t$

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

In a right triangle, the lengths of the two smallest sides are 5 and 12. Find the length of the hypotenuse.

Possible Answers:

Correct answer:

Explanation:

In order to find the length of the hypotenuse, we need to use the pythagorean theorem, which states that

a^2 + b^2 = c^2

By substituting 5 for a and 12 for b, we get

5^2 + 12^2 = c^2

or,

169 = c^2

To solve for c we need to take the square root of 169, which is 13. Therefore, the hypotenuse is 13.

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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:

100

200

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 #131 : Plane Geometry

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 #112 : Geometry

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:

9, 12, 15

6,7,12

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

5, 12, 13

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 #2 : How To Find The Length Of The Side Of A Right Triangle

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

Possible Answers:

171

14.87

13.07

Correct answer:

13.07

Explanation:

The Pythagorean Theorem gives us a² + b² = c² for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so b² = 14² – 5² = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

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

Which of the following could NOT be the lengths of the sides of a right triangle?

Possible Answers:

14, 48, 50

5, 7, 10

12, 16, 20

8, 15, 17

5, 12, 13

Correct answer:

5, 7, 10

Explanation:

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem a² + b² = c²

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Which set of sides could make a right triangle?

Possible Answers:

6, 7, 8

10, 12, 16

9, 12, 15

4, 6, 9

Correct answer:

9, 12, 15

Explanation:

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

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

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

Screen_shot_2013-03-18_at_10.29.39_pm

Possible Answers:

5.5

4.5

3.5

Correct answer:

Explanation:

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗²–〖12〗²) = 9, so x=9 – 5=4

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

Possible Answers:

12 √6

33√2

Correct answer:

Explanation:

use the pythagorean theorem:

a² + b² = c² ; a and b are sides, c is the hypotenuse

a² + 1296 = 1521

a² = 225

a = 15

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SAT Math : Triangles

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #94 : Geometry

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 #131 : Plane Geometry

Example Question #112 : Geometry

Example Question #2 : 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

Example Question #2 : 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

Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

All SAT Math Resources

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