SAT Math : Geometry

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #62 : Triangles

Figure6

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

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

Figure7

Find the hypotenuse of the right triangle with the Pythagorean Theorem, , where  and  are the legs of the triangle and  is its hypotenuse. 

This is our missing length.

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

Example Question #61 : 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 #81 : Triangles

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:

Quantity A is greater.

 

The two quantities are equal.

 

Quantity B is greater.

 

The relationship cannot be determined from the information provided.

 

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 #91 : 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:

Correct answer:

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 

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 .

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

Example Question #13 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

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 #92 : Plane Geometry

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:

Correct answer:

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 ).

 

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.

Example Question #71 : Triangles

Square 1

The above figure shows Square  is the midpoint of  is the midpoint of . Construct 

. Which of the following expresses the length of   in terms of ?

Possible Answers:

Correct answer:

Explanation:

Since all four sides of a square are congruent, 

Since  is the midpoint of ,

Since  is the midpoint of 

,

and 

 is a right triangle, so, by the Pythagorean Theorem, 

Substituting: 

Apply the Product of Radicals and Quotient of Radicals Rules:

Example Question #94 : Plane Geometry

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

 

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

or,

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

Example Question #81 : Triangles

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

50

70

25

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  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 #82 : Triangles

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

19

23

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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