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SAT Math : Plane Geometry

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Example Questions

Example Question #62 : Triangles

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

Divide the shape into a rectangle and a right triangle as indicated below.

Find the hypotenuse of the right triangle with the Pythagorean Theorem, a^2 + b^2 = c^2 , where and are the legs of the triangle and is its hypotenuse.

(8)^2+(6)^2 = c^2

100 = c^2

$\sqrt{100} = \sqrt{c^2}$

c =10

This is our missing length.

Now add the sides of the polygon together to find the perimeter:

20 + 10 + 26 + 8 = 64

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Example Question #63 : 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:

(x+2)-(x+1)=x

x+x-3=x

x^2+(x+1)^2=(x+2)^2

x^2+(x+2)^2=(x+4)^2

(x)(x+1)=(x+2)^2

Correct answer:

x^2+(x+1)^2=(x+2)^2

Explanation:

Since the lengths of the sides are consecutive integers and the shortest side is , the three sides are , (x+1) , and (x+2) .

We then use the Pythagorean Theorem:

$\newline a^2+b^2=c^2 \newline x^2+(x+1)^2=(x+2)^2$

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Example Question #114 : Geometry

Square PQRS is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point has coordinates $\left ( -2,-1 \right )$ and point has the coordinates $\left ( 3,4 \right )$ .

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 $s:s:s\sqrt{2}$ , so if the sides have a length of 5, the hypotenuse must be $5\sqrt{2}$ .

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Example Question #94 : Plane Geometry

Two sides of a given triangle are both . If one angle of the triangle is a right angle, then what is the measure of the hypotenuse?

Possible Answers:

$5\sqrt{2}$

$7\sqrt{5}$

$2\sqrt{5}$

Correct answer:

$5\sqrt{2}$

Explanation:

If we know two sides are equal to and we know that one of the angles is a right angle, then that means that this must be a Special Right Triangle where the interior angles are $90$^{\circ}$, 45$^{\circ}$, 45$^{\circ}$$ .

With this special triangle, we also know that the measure of the hypotenuse is equal to the measure of one side of the triangle times the square root of that measure.

Since one leg of the triangle is . then the hypotenuse is equal to $5\sqrt{2}$ .

We could also solve this using the Pythagorean Theorem, like so:

$c^2=\sqrt{5^2+5^2}=\sqrt{50}=\sqrt{2\cdot25}=5\sqrt{2}$

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Example Question #194 : Plane Geometry

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:

$25\textup{ ft}$

$35\textup{ ft}$

$30\textup{ ft}$

$45\textup{ ft}$

$22\textup{ ft}$

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

If a 18-foot light pole casts a shadow of 7 feet on even ground, what is the distance from the top of the light pole to the top of it’s shadow?

Possible Answers:

19.31 ft

17.78 ft

25.12 ft

18.36 ft

21.00 ft

Correct answer:

19.31 ft

Explanation:

First, you will want to draw a small diagram to help you:

Lightpole example

After looking at the diagram, it is clear this is a right triangle problem where we can use the Pythagorean Theorem (a² + b² = c² ).

7^2 + 18^2 = c^2

49 + 324= c^2

373 = c^2

$\displaystyle \sqrt{373}= 19.31$

The distance from the top of the light pole to the top of it’s shadow is 19.31 feet.

Process of Elimination Hint: You can immediate eliminate 2 answers based on the properties of right triangles. The hypotenuse has to be longer than either one of the other two sides (must be greater than 18 ft) and less than the length of the other two sides added together (must be less than 25 feet). Therefore, 17.78 feet and 25.12 feet are unreasonable answers.

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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 #195 : Plane Geometry

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 #132 : Act Math

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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SAT Math : Plane Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #62 : Triangles

Example Question #63 : Triangles

Example Question #114 : Geometry

Example Question #94 : Plane Geometry

Example Question #194 : Plane Geometry

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

Example Question #94 : Geometry

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

Example Question #195 : Plane Geometry

Example Question #132 : Act Math

All SAT Math Resources

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