SAT Math : Plane Geometry

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #33 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Triangle

If the length of CB is 6 and the angle C measures 45º, what is the length of AC in the given right triangle?

Possible Answers:

72

9

6√2

6

12√2

Correct answer:

6√2

Explanation:

Pythagorean Theorum

AB2 + BC2 = AC2

If C is 45º then A is 45º, therefore AB = BC

AB2 + BC2 = AC2

62 + 62 = AC2

2*62 = AC2

AC = √(2*62) = 6√2

Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

You leave on a road trip driving due North from Savannah, Georgia, at 8am.  You drive for 5 hours at 60mph and then head due East for 2 hours at 50mph.  After those 7 hours, how far are you Northeast from Savannah as the crow flies (in miles)?

Possible Answers:

Correct answer:

Explanation:

Distance = hours * mph

North Distance = 5 hours * 60 mph = 300 miles

East Distance = 2 hours * 50 mph = 100 miles

Use Pythagorean Theorem to determine Northeast Distance

3002 + 1002 =NE2

90000  + 10000 = 100000 = NE2

NE = √100000

Example Question #34 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A square garden has an area of 49 ft2. To the nearest foot, what is the diagonal distance across the garden?

Possible Answers:

8

10

7

11

9

Correct answer:

10

Explanation:

Since the garden is square, the two sides are equal to the square root of the area, making each side 7 feet. Then, using the Pythagorean Theorem, set up the equation 7+ 7= the length of the diagonal squared. The length of the diagonal is the square root of 98, which is closest to 10.

Example Question #82 : Sat Mathematics

A man at the top of a lighthouse is watching birds through a telescope. He spots a pelican 5 miles due north of the lighthouse. The pelican flies due west for 12 miles before resting on a buoy. What is the distance, in miles, from the pelican's current resting spot to the lighthouse?

Possible Answers:

Correct answer:

Explanation:

We look at the 3 points of interest: the lighthouse, where the pelican started, and where the pelican ended. We can see that if we connect these 3 points with lines, they form a right triangle. (From due north, flying exactly west creates a 90 degree angle.) The three sides of the triangle are 5 miles, 13 miles and an unknown distance. Using the Pythagorean Theorem we get:

Example Question #1251 : Basic Geometry

An airplane is 8 miles west and 15 miles south of its destination.  Approximately how far is the plane from its destination, in miles?

 

 

Possible Answers:

Correct answer:

Explanation:

A right triangle can be drawn between the airplane and its destination.

                           Destination

                      15 miles  Act_math_170_01  Airplane

                                     8 miles

We can solve for the hypotenuse, x, of the triangle:

82 + 152 = x2

64 + 225 = x2

289 = x2

x = 17 miles

 

 

Example Question #84 : Sat Mathematics

An 8-foot-tall tree is perpendicular to the ground and casts a 6-foot shadow. What is the distance, to the nearest foot, from the top of the tree to the end of the shadow? 

Possible Answers:

\dpi{100} \small 8

\dpi{100} \small 4

\dpi{100} \small 6

\dpi{100} \small 10

\dpi{100} \small 5

Correct answer:

\dpi{100} \small 10

Explanation:

In order to find the distance from the top of the tree to the end of the shadow, draw a right triangle with the height(tree) labeled as 8 and base(shadow) labeled as 6: 

 Screen_shot_2013-08-16_at_12.34.40_am

From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:

\dpi{100} \small a^{2}+b^{2}=c^{2} 

or you can notice that this is simililar to a 3-4-5 triangle. Since the lengths are just increased by a factor of 2, the hypotenuse that is normally 5 would be 10. 

Example Question #85 : Sat Mathematics

Screen_shot_2013-03-18_at_10.21.29_pm

 

In the figure above,  is a square and  is three times the length of . What is the area of ?

Possible Answers:

Correct answer:

Explanation:

Assigning the length of ED the value of x, the value of AE will be 3x. That makes the entire side AD equal to 4x. Since the figure is a square, all four sides will be equal to 4x. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3x, 4x and 10. Using the Pythagorean theorem:

(3x)2 + (4x)2 = 102

9x2 + 16x2 = 100

25x2 = 100

x2 = 4

x = 2

With x = 2, each side of the square is 4x, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.

Example Question #81 : Sat Mathematics

1

The hypotenuse is the diameter of the circle. Find the area of the circle above.

Possible Answers:

6.25\pi

6.75\pi

5\pi

6.5\pi

5.5\pi

Correct answer:

6.25\pi

Explanation:

Using the Pythagorean Theorem, we can find the length of the hypotenuse:

3^{2}+4^{2}=5^{2}.

Therefore the hypotenuse has length 5.

The area of the circle is \pi r^{2}=\pi \cdot 2.5^{2}=6.25\pi

Example Question #61 : Triangles

Find the length of the hypotenuse.

Triangle_4_14_c

Note: This is a right triangle.

Possible Answers:

Correct answer:

Explanation:

To find the length of this hypotenuse, we need to use the Pythagorean Theorem:

, where a and b are the legs and c is the hypotenuse.

Here, c is our missing hypotenuse length, a = 4 ,and b = 14.

Plug these values in and solve for c:

 

 

 

Example Question #88 : Sat Mathematics

Side  in the triangle below (not to scale) is equal to . Side  is equal to . What is the length of side ?

Right_triangle_with_labeled_sides

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Theorem: , where a and b are the legs and c is the hypotenuse.

We know  and , so we can plug them in to solve for c:

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