All PSAT Math Resources
Example Questions
Example Question #113 : 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?
A right triangle can be drawn between the airplane and its destination.
Destination
15 miles 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 #111 : Geometry
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?
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:
From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:
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 #112 : Geometry
In the figure above, is a square and is three times the length of . What is the area of ?
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 #121 : Geometry
The hypotenuse is the diameter of the circle. Find the area of the circle above.
Using the Pythagorean Theorem, we can find the length of the hypotenuse:
.
Therefore the hypotenuse has length 5.
The area of the circle is
Example Question #122 : Geometry
Find the length of the hypotenuse.
Note: This is a right triangle.
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 #123 : Geometry
Side in the triangle below (not to scale) is equal to . Side is equal to . What is the length of side ?
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:
Example Question #124 : Geometry
Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?
9.43 miles
13 miles
89 miles
4.36 miles
19 miles
9.43 miles
We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.
Apply the Pythagorean Theorem:
a2 + b2 = c2
25 + 64 = c2
89 = c2
c = 9.43 miles
Example Question #125 : Geometry
What is the hypotenuse of a right triangle with side lengths and ?
The Pythagorean Theorem states that . This question gives us the values of and , and asks us to solve for .
Take and and plug them into the equation as and :
Now we can start solving for :
The length of the hypotenuse is .
Example Question #126 : Geometry
One leg of a triangle measures 12 inches. Which of the following could be the length of the other leg if the hypotenuse is an integer length?
By the Pythagorean Theorem, if is the hypotenuse and and are the legs, .
Set , the known leg, and rewrite the above as:
We can now substitute each of the five choices for ; the one which yields a whole number for is the correct answer choice.
:
:
:
:
:
The only value of which yields a whole number for the hypotenuse is 16, so this is the one we choose.
Example Question #127 : Geometry
Find the perimeter of the polygon.
Divide the shape into a rectangle and a right triangle as indicated below.
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: