How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem - Math
Card 1 of 724
What is the hypotenuse of a right triangle with sides 5 and 8?
What is the hypotenuse of a right triangle with sides 5 and 8?
Tap to reveal answer
Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
_a_2 + _b_2 = _c_2
52 + 82 = _c_2
25 + 64 = _c_2
89 = _c_2
c = √89
Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
_a_2 + _b_2 = _c_2
52 + 82 = _c_2
25 + 64 = _c_2
89 = _c_2
c = √89
← Didn't Know|Knew It →
Which is the greater quantity?
(a) The hypotenuse of a right triangle with legs 
and 
.
(b) The hypotenuse of a right triangle with legs 
and 
.
Which is the greater quantity?
(a) The hypotenuse of a right triangle with legs
(b) The hypotenuse of a right triangle with legs
Tap to reveal answer
The hypotenuses of the triangles measure as follows:
(a) 
(b) 
, so 
, making (a) the greater quantity.
The hypotenuses of the triangles measure as follows:
(a)
(b)
← Didn't Know|Knew It →
Which is the greater quantity?
(a) The hypotenuse of a 
right triangle with a leg of length 20
(b) The hypotenuse of a right triangle with legs of length 19 and 21
Which is the greater quantity?
(a) The hypotenuse of a
(b) The hypotenuse of a right triangle with legs of length 19 and 21
Tap to reveal answer
The hypotenuses of the triangles measure as follows:
(a) 
(b) 
, so 
, making (b) the greater quantity
The hypotenuses of the triangles measure as follows:
(a)
(b)
← Didn't Know|Knew It →
A right triangle has a leg 
feet long and a hypotenuse 
feet long. Which is the greater quantity?
(a) The length of the second leg of the triangle
(b) 60 inches
A right triangle has a leg
(a) The length of the second leg of the triangle
(b) 60 inches
Tap to reveal answer
The length of the second leg can be calculated using the Pythagorean Theorem. Set 
:
The second leg therefore measures 
inches.
The length of the second leg can be calculated using the Pythagorean Theorem. Set
The second leg therefore measures
← Didn't Know|Knew It →
What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?
What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?
Tap to reveal answer
Since we're dealing with right triangles, we can use the Pythagorean Theorem (
). In this formula, a and b are the sides, while c is the hypotenuse. The hypotenuse of a right triangle is the longest side and the side that is opposite the right angle. Now, we can plug into our formula, which looks like this: 
We simplify and get 
. At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.
Since we're dealing with right triangles, we can use the Pythagorean Theorem (
← Didn't Know|Knew It →
The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?
(A) The length of one side of the pentagon
(B) One and one-half feet
The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?
(A) The length of one side of the pentagon
(B) One and one-half feet
Tap to reveal answer
By the Pythagorean Theorem, the hypotenuse of the right triangle is
inches, making its perimeter
inches.
The pentagon in question has sides of length 75% of 112, or
.
Since a pentagon has five sides of equal length, each side will have measure
inches.
One and a half feet are equivalent to 
inches, so (B) is the greater quantity.
By the Pythagorean Theorem, the hypotenuse of the right triangle is
The pentagon in question has sides of length 75% of 112, or
Since a pentagon has five sides of equal length, each side will have measure
One and a half feet are equivalent to
← Didn't Know|Knew It →
The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.
Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.
Which is the greater quantity?
(A) The distance Cary runs
(B) One-fourth of a mile
The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.
Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.
Which is the greater quantity?
(A) The distance Cary runs
(B) One-fourth of a mile
Tap to reveal answer
By the Pythagorean Theorem, the distance from B to C is
feet
Cary runs
feet
Since 5,280 feet make a mile, one-fourth of a mile is equal to
feet.
(B) is greater
By the Pythagorean Theorem, the distance from B to C is
Cary runs
Since 5,280 feet make a mile, one-fourth of a mile is equal to
(B) is greater
← Didn't Know|Knew It →
Give the length of the hypotenuse of the above right triangle in terms of 
.
Give the length of the hypotenuse of the above right triangle in terms of
Tap to reveal answer
If we let 
be the length of the hypotenuse, then by the Pythagorean theorem,
If we let
← Didn't Know|Knew It →
In Square 
. 
is the midpoint of 
, 
is the midpoint of 
, and 
is the midpoint of 
. Construct the line segments 
and 
.
Which is the greater quantity?
(a) 
(b) 
In Square
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
The figure referenced is below:
For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.
and 
are midpoints of their respective sides, so 
, making 
the hypotenuse of a triangle with legs of length 2 and 2. Therefore,
.
Also, 
, and since 
is the midpoint of 
, 
. 
, making 
the hypotenuse of a triangle with legs of length 1 and 4. Therefore,
, so 
The figure referenced is below:
For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.
Also,
← Didn't Know|Knew It →
Figure NOT drawn to scale.
In the above figure, 
is a right angle.
What is the length of 
?
Figure NOT drawn to scale.
In the above figure,
What is the length of
Tap to reveal answer
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
.
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
← Didn't Know|Knew It →
Figure NOT drawn to scale.
In the above figure, 
is a right angle.
What is the length of 
?
Figure NOT drawn to scale.
In the above figure,
What is the length of
Tap to reveal answer
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
.
Their corresponding sides are in proportion, so, setting the ratios of the long legs to the short legs equal to each other,
By the Pythagorean Theorem.
The proportion statement becomes
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the long legs to the short legs equal to each other,
By the Pythagorean Theorem.
The proportion statement becomes
← Didn't Know|Knew It →
Given: 
with 
, 
, 
.
Which is the greater quantity?
(a) 
(b) 
Given:
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:
; it follows that 
is obtuse, and has measure greater than 
The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:
← Didn't Know|Knew It →
Figure NOT drawn to scale.
In the above figure, is a right angle.
What is the length of ?
Figure NOT drawn to scale.
In the above figure,
What is the length of
Tap to reveal answer
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
.
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular,
Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,
← Didn't Know|Knew It →
Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?
Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?
Tap to reveal answer
First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(12 + 12) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.
First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(12 + 12) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.
← Didn't Know|Knew It →
Which of the following sets of sides cannnot belong to a right triangle?
Which of the following sets of sides cannnot belong to a right triangle?
Tap to reveal answer
To answer this question without plugging all five answer choices in to the Pythagorean Theorem (which takes too long on the GRE), we can use special triangle formulas. Remember that 45-45-90 triangles have lengths of x, x, x√2. Similarly, 30-60-90 triangles have lengths x, x√3, 2x. We should also recall that 3,4,5 and 5,12,13 are special right triangles. Therefore the set of sides that doesn't fit any of these rules is 6, 7, 8.
To answer this question without plugging all five answer choices in to the Pythagorean Theorem (which takes too long on the GRE), we can use special triangle formulas. Remember that 45-45-90 triangles have lengths of x, x, x√2. Similarly, 30-60-90 triangles have lengths x, x√3, 2x. We should also recall that 3,4,5 and 5,12,13 are special right triangles. Therefore the set of sides that doesn't fit any of these rules is 6, 7, 8.
← Didn't Know|Knew It →
Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?
Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?
Tap to reveal answer
This can be solved with the Pythagorean Theorem.
62 + 42 = _c_2
52 = _c_2
c = √52 = 2√13
This can be solved with the Pythagorean Theorem.
62 + 42 = _c_2
52 = _c_2
c = √52 = 2√13
← Didn't Know|Knew It →
Paul leaves his home and jogs 3 miles due north and 4 miles due west. If Paul could walk a straight line from his current position back to his house, how far, in miles, is Paul from home?
Paul leaves his home and jogs 3 miles due north and 4 miles due west. If Paul could walk a straight line from his current position back to his house, how far, in miles, is Paul from home?
Tap to reveal answer
By using the Pythagorean Theorem, we can solve for the distance “as the crow flies” from Paul to his home:
32 + 42 = _x_2
9 + 16 = _x_2
25 = _x_2
5 = x
By using the Pythagorean Theorem, we can solve for the distance “as the crow flies” from Paul to his home:
32 + 42 = _x_2
9 + 16 = _x_2
25 = _x_2
5 = x
← Didn't Know|Knew It →
Which set of side lengths CANNOT correspond to a right triangle?
Which set of side lengths CANNOT correspond to a right triangle?
Tap to reveal answer
Because we are told this is a right triangle, we can use the Pythagorean Theorem, _a_2 + _b_2 = _c_2. You may also remember some of these as special right triangles that are good to memorize, such as 3, 4, 5.
Here, 6, 8, 11 will not be the sides to a right triangle because 62 + 82 = 102.
Because we are told this is a right triangle, we can use the Pythagorean Theorem, _a_2 + _b_2 = _c_2. You may also remember some of these as special right triangles that are good to memorize, such as 3, 4, 5.
Here, 6, 8, 11 will not be the sides to a right triangle because 62 + 82 = 102.
← Didn't Know|Knew It →
Given a right triangle where the two legs have lengths of 3 and 4 respectively, what is the length of the hypotenuse?
Given a right triangle where the two legs have lengths of 3 and 4 respectively, what is the length of the hypotenuse?
Tap to reveal answer
The hypotenuse can be found using Pythagorean Theorem, which is a2 + b2 = c2, so we plug in a = 3 and b = 4 to get c.
c2 =25, so c = 5
The hypotenuse can be found using Pythagorean Theorem, which is a2 + b2 = c2, so we plug in a = 3 and b = 4 to get c.
c2 =25, so c = 5
← Didn't Know|Knew It →
What is the hypotenuse of a right triangle with sides 5 and 8?
What is the hypotenuse of a right triangle with sides 5 and 8?
Tap to reveal answer
Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
_a_2 + _b_2 = _c_2
52 + 82 = _c_2
25 + 64 = _c_2
89 = _c_2
c = √89
Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.
_a_2 + _b_2 = _c_2
52 + 82 = _c_2
25 + 64 = _c_2
89 = _c_2
c = √89
← Didn't Know|Knew It →