How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem - PSAT Math
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In the figure above, 
is a square and 
is three times the length of 
. What is the area of 
?
In the figure above,
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Assigning the length of ED the value of x, the value of AE will be 3_x_. That makes the entire side AD equal to 4_x_. Since the figure is a square, all four sides will be equal to 4_x_. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3_x_, 4_x_ and 10. Using the Pythagorean theorem:
(3_x_)2 + (4_x_)2 = 102
9_x_2 + 16_x_2 = 100
25_x_2 = 100
_x_2 = 4
x = 2
With x = 2, each side of the square is 4_x_, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.
Assigning the length of ED the value of x, the value of AE will be 3_x_. That makes the entire side AD equal to 4_x_. Since the figure is a square, all four sides will be equal to 4_x_. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3_x_, 4_x_ and 10. Using the Pythagorean theorem:
(3_x_)2 + (4_x_)2 = 102
9_x_2 + 16_x_2 = 100
25_x_2 = 100
_x_2 = 4
x = 2
With x = 2, each side of the square is 4_x_, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.
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In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?
In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?
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Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100√2=c. 10√2=c
Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100√2=c. 10√2=c
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Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?
Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?
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By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:
102 + 52 = _x_2
100 + 25 = _x_2
√125 = x, but we still need to factor the square root
√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so
5√5= x
By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:
102 + 52 = _x_2
100 + 25 = _x_2
√125 = x, but we still need to factor the square root
√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so
5√5= x
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If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?
If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?
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Using the Pythagorean theorem, _x_2 + _y_2 = _h_2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = _h_2 . Multiplied out 25 + 25 = _h_2.
Therefore _h_2 = 50, so h = √50 = √2 * √25 or 5√2.
Using the Pythagorean theorem, _x_2 + _y_2 = _h_2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = _h_2 . Multiplied out 25 + 25 = _h_2.
Therefore _h_2 = 50, so h = √50 = √2 * √25 or 5√2.
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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?
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?
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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.
First, define the sides of the triangle. Because the side lengths are consecutive odd numbers, if we define the shortest side will be as 
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 
The longest side of the triangle is 21 centimeters long.
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Angela drives 30 miles north and then 40 miles east. How far is she from where she began?
Angela drives 30 miles north and then 40 miles east. How far is she from where she began?
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By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a2+b2=c2, 302+402=c2, c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.
By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a2+b2=c2, 302+402=c2, c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.
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An airplane is 8 miles west and 15 miles south of its destination. Approximately how far is the plane from its destination, in miles?
An airplane is 8 miles west and 15 miles south of its destination. Approximately how far is the plane from its destination, in miles?
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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
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
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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 
?
The lengths of the sides of a right triangle are consecutive integers, and the length of the shortest side is
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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:
Since the lengths of the sides are consecutive integers and the shortest side is
We then use the Pythagorean Theorem:
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Each of the following answer choices lists the side lengths of a different triangle. Which of these triangles does not have a right angle?
Each of the following answer choices lists the side lengths of a different triangle. Which of these triangles does not have a right angle?
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cannot be the side lengths of a right triangle. 
does not equal 
. Also, special right triangle 
and 
rules can eliminate all the other choices.
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The height of a right circular cylinder is 10 inches and the diameter of its base is 6 inches. What is the distance from a point on the edge of the base to the center of the entire cylinder?
The height of a right circular cylinder is 10 inches and the diameter of its base is 6 inches. What is the distance from a point on the edge of the base to the center of the entire cylinder?
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The best thing to do here is to draw diagram and draw the appropiate triangle for what is being asked. It does not matter where you place your point on the base because any point will produce the same result. We know that the center of the base of the cylinder is 3 inches away from the base (6/2). We also know that the center of the cylinder is 5 inches from the base of the cylinder (10/2). So we have a right triangle with a height of 5 inches and a base of 3 inches. So using the Pythagorean Theorem 32 + 52 = c2. 34 = c2, c = √(34).
The best thing to do here is to draw diagram and draw the appropiate triangle for what is being asked. It does not matter where you place your point on the base because any point will produce the same result. We know that the center of the base of the cylinder is 3 inches away from the base (6/2). We also know that the center of the cylinder is 5 inches from the base of the cylinder (10/2). So we have a right triangle with a height of 5 inches and a base of 3 inches. So using the Pythagorean Theorem 32 + 52 = c2. 34 = c2, c = √(34).
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If the length of CB is 6 and the angle C measures 45º, what is the length of AC in the given right 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?
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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
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
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A square enclosure has a total area of 3,600 square feet. What is the length, in feet, of a diagonal across the field rounded to the nearest whole number?
A square enclosure has a total area of 3,600 square feet. What is the length, in feet, of a diagonal across the field rounded to the nearest whole number?
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In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.
The area of a square is found by multiply the lengths of 2 sides of a square by itself.
So, the square root of 3,600 comes out to 60 ft.
The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.
602 + 602 = C2
the square root of 7,200 is 84.8, which can be rounded to 85
In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.
The area of a square is found by multiply the lengths of 2 sides of a square by itself.
So, the square root of 3,600 comes out to 60 ft.
The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.
602 + 602 = C2
the square root of 7,200 is 84.8, which can be rounded to 85
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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?
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?
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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
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
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To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?
To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?
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Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.
Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.
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Justin travels 
to the east and 
to the north. How far away from his starting point is he now?
Justin travels
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This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that 
This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that
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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)?
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)?
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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
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
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A right triangle with sides A, B, C and respective angles a, b, c has the following measurements.
Side A = 3in. Side B = 4in. What is the length of side C?
A right triangle with sides A, B, C and respective angles a, b, c has the following measurements.
Side A = 3in. Side B = 4in. What is the length of side C?
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The correct answer is 5. The pythagorean theorem states that a2 + b2 = c2. So in this case 32 + 42 = C2. So C2 = 25 and C = 5. This is also an example of the common 3-4-5 triangle.
The correct answer is 5. The pythagorean theorem states that a2 + b2 = c2. So in this case 32 + 42 = C2. So C2 = 25 and C = 5. This is also an example of the common 3-4-5 triangle.
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A park is designed to fit within the confines of a triangular lot in the middle of a city. The side that borders Elm street is 15 feet long. The side that borders Broad street is 23 feet long. Elm street and Broad street meet at a right angle. The third side of the park borders Popeye street, what is the length of the side of the park that borders Popeye street?
A park is designed to fit within the confines of a triangular lot in the middle of a city. The side that borders Elm street is 15 feet long. The side that borders Broad street is 23 feet long. Elm street and Broad street meet at a right angle. The third side of the park borders Popeye street, what is the length of the side of the park that borders Popeye street?
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This question requires the use of Pythagorean Theorem. We are given the length of two sides of a triangle and asked to find the third. We are told that the two sides we are given meet at a right angle, this means that the missing side is the hypotenuse. So we use a2 + b2 = c2, plugging in the two known lengths for a and b. This yields an answer of 27.46 feet.
This question requires the use of Pythagorean Theorem. We are given the length of two sides of a triangle and asked to find the third. We are told that the two sides we are given meet at a right angle, this means that the missing side is the hypotenuse. So we use a2 + b2 = c2, plugging in the two known lengths for a and b. This yields an answer of 27.46 feet.
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A square garden has an area of 49 ft2. To the nearest foot, what is the diagonal distance across the garden?
A square garden has an area of 49 ft2. To the nearest foot, what is the diagonal distance across the garden?
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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 72 + 72 = the length of the diagonal squared. The length of the diagonal is the square root of 98, which is closest to 10.
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 72 + 72 = the length of the diagonal squared. The length of the diagonal is the square root of 98, which is closest to 10.
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The lengths of the three sides of a right triangle form a set of consecutive even integers when arranged from least to greatest. If the second largest side has a length of x, then which of the following equations could be used to solve for x?
The lengths of the three sides of a right triangle form a set of consecutive even integers when arranged from least to greatest. If the second largest side has a length of x, then which of the following equations could be used to solve for x?
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We are told that the lengths form a series of consecutive even integers. Because even integers are two units apart, the side lengths must differ by two. In other words, the largest side length is two greater than the second largest, and the second largest length is two greater than the smallest length.
The second largest length is equal to x. The second largest length must thus be two less than the largest length. We could represent the largest length as x + 2.
Similarly, the second largest length is two larger than the smallest length, which we could thus represent as x – 2.
To summarize, the lengths of the triangle (in terms of x) are x – 2, x, and x + 2.
In order to solve for x, we can make use of the fact that the triangle is a right triangle. If we apply the Pythagorean Theorem, we can set up an equation that could be used to solve for x. The Pythagorean Theorem states that if a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse, then the following is true:
a2 + b2 = c2
In this particular case, the two legs of our triangle are x – 2 and x, since the legs are the two smallest sides; therefore, we can say that a = x – 2, and b = x. Lastly, we can say c = x + 2, since x + 2 is the length of the hypotenuse. Subsituting these values for a, b, and c into the Pythagorean Theorem yields the following:
(x – 2)2 + x2 = (x + 2)2
The answer is (x – 2)2 + x2 = (x + 2)2.
We are told that the lengths form a series of consecutive even integers. Because even integers are two units apart, the side lengths must differ by two. In other words, the largest side length is two greater than the second largest, and the second largest length is two greater than the smallest length.
The second largest length is equal to x. The second largest length must thus be two less than the largest length. We could represent the largest length as x + 2.
Similarly, the second largest length is two larger than the smallest length, which we could thus represent as x – 2.
To summarize, the lengths of the triangle (in terms of x) are x – 2, x, and x + 2.
In order to solve for x, we can make use of the fact that the triangle is a right triangle. If we apply the Pythagorean Theorem, we can set up an equation that could be used to solve for x. The Pythagorean Theorem states that if a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse, then the following is true:
a2 + b2 = c2
In this particular case, the two legs of our triangle are x – 2 and x, since the legs are the two smallest sides; therefore, we can say that a = x – 2, and b = x. Lastly, we can say c = x + 2, since x + 2 is the length of the hypotenuse. Subsituting these values for a, b, and c into the Pythagorean Theorem yields the following:
(x – 2)2 + x2 = (x + 2)2
The answer is (x – 2)2 + x2 = (x + 2)2.
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