How to find the length of the hypotenuse of a 45/45/90 right isosceles triangle : Pythagorean Theorem - Geometry
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What is the length of the diagonal of a square with sides of 
?
What is the length of the diagonal of a square with sides of
The diagonal cuts a square into two 
right triangles with the legs of the triangle being the sides of the square and the hypotenuse is the diagonal of the triangle.
Using Pythagorean Theorem we get:
or 
The diagonal cuts a square into two
Using Pythagorean Theorem we get:
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Compare your answer with the correct one above
A right triangle has two sides of length 
; what is the correct formula for finding the length of the hypotenuse 
?
A right triangle has two sides of length
The Pythagorean Theorem states that the sum of the squares of the sides of the triangle are equal to the square of the hypotenuse, summed up as 
but for the case of right isoceles triangles where a and b are equal, it can be rewritten as:
,
which simplifies to
.
The Pythagorean Theorem states that the sum of the squares of the sides of the triangle are equal to the square of the hypotenuse, summed up as
which simplifies to
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is a 
triangle.
Using the Pythagorean Theorem, calculate the length of the hypotenuse of .
Using the Pythagorean Theorem, calculate the length of the hypotenuse of
The Pythagorean Theorem holds that, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. That is,
where 
is the hypotenuse and 
and 
are the other two sides.
Here, 
corresponds to 
, and, since this is a 
triangle, the length of 
equals the length of 
, so we know that 
.
Apply the Pythagorean Theorem.
So the length of the hypotenuse is 
.
The Pythagorean Theorem holds that, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. That is,
where
Here,
Apply the Pythagorean Theorem.
So the length of the hypotenuse is
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The following image is not to scale.
Find the length of the hypotenuse of the right triangle:
The following image is not to scale.
Find the length of the hypotenuse of the right triangle:
Because the problem states that this the triangle is a right triangle and provides the length of the two legs, the third side (hypotenuse) can be calculated using the Pythagorean Theorem.
, where a and b can be arbitrarily chosen for either leg length and c represents the length of the hypotenuse.
but because the question requires to find the length of the hypotenuse, c cannot be a negative number. Therefore the final answer is +13.9.
Because the problem states that this the triangle is a right triangle and provides the length of the two legs, the third side (hypotenuse) can be calculated using the Pythagorean Theorem.
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The following image is not to scale.
Find the length of the hypotenuse of the 
right triangle.
The following image is not to scale.
Find the length of the hypotenuse of the
45/45/90 triangles are always isosceles. This means that two of the legs of the triangle are congruent. In the figure, it's indicates which two sides are congruent.
From here, we can find the length of the hypotenuse through the Pythagorean Theorem. We can confirm this because the problem has given us no angle measures to perform trig functions with.
c also equals -2√2, but because the hypotenuse is a physical length, the value cannot be negative distance.
45/45/90 triangles are always isosceles. This means that two of the legs of the triangle are congruent. In the figure, it's indicates which two sides are congruent.
From here, we can find the length of the hypotenuse through the Pythagorean Theorem. We can confirm this because the problem has given us no angle measures to perform trig functions with.
c also equals -2√2, but because the hypotenuse is a physical length, the value cannot be negative distance.
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Square 
has a side length of 
. What is the length of its diagonal?
Square
The answer can be found two different ways. The first step is to realize that this is really a triangle question, even though it starts with a square. By drawing the square out and adding the diagonal, you can see that you form two right triangles. Furthermore, the diagonal bisects two ninety-degree angles, thereby making the resulting triangles a 
triangle.
From here you can go one of two ways: using the Pythagorean Theorem to find the diagonal, or recognizing the triangle as a 
triangle.
- Using the Pythagorean Theorem
Once you recognize the right triangle in this question, you can begin to use the Pythagorean Theorem. Remember the formula: 
, where 
and 
are the lengths of the legs of the triangle, and 
is the length of the triangle's hypotenuse.
In this case, 
. We can substitute these values into the equation and then solve for 
, the hypotenuse of the triangle and the diagonal of the square:
The length of the diagonal is 
.
- Using Properties of
Triangles
The second approach relies on recognizing a 
triangle. Although one could solve this rather easily with Pythagorean Theorem, the following method could be faster.
triangles have side length ratios of 
, where 
represents the side lengths of the triangle's legs and 
represents the length of the hypotenuse.
In this case, 
because it is the side length of our square and the triangles formed by the square's diagonal.
Therefore, using the 
triangle ratios, we have 
for the hypotenuse of our triangle, which is also the diagonal of our square.
The answer can be found two different ways. The first step is to realize that this is really a triangle question, even though it starts with a square. By drawing the square out and adding the diagonal, you can see that you form two right triangles. Furthermore, the diagonal bisects two ninety-degree angles, thereby making the resulting triangles a
From here you can go one of two ways: using the Pythagorean Theorem to find the diagonal, or recognizing the triangle as a
- Using the Pythagorean Theorem
Once you recognize the right triangle in this question, you can begin to use the Pythagorean Theorem. Remember the formula:
In this case,
The length of the diagonal is
- Using Properties of
The second approach relies on recognizing a
In this case,
Therefore, using the
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What is the length of the hypotenuse of an isosceles right triangle with an area of 
?
What is the length of the hypotenuse of an isosceles right triangle with an area of
Recall that an isosceles right triangle is also a 
triangle. It has sides that appear as follows:
Therefore, the area of the triangle is:
, since the base and the height are the same.
For our data, this means:
Solving for 
, you get:
So, your triangle looks like this:
Now, you can solve this with a ratio and easily find that it is 
. You also can use the Pythagorean Theorem. To do the latter, it is:
Now, just do your math carefully:
That is a weird kind of factoring, but it makes sense if you distribute back into the group. This means you can simplify:
Recall that an isosceles right triangle is also a
Therefore, the area of the triangle is:
For our data, this means:
Solving for
So, your triangle looks like this:
Now, you can solve this with a ratio and easily find that it is
Now, just do your math carefully:
That is a weird kind of factoring, but it makes sense if you distribute back into the group. This means you can simplify:
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When the sun shines on a 
pole, it leaves a shadow on the ground that is also 
. What is the distance from the top of the pole to the end of its shadow?
When the sun shines on a
The pole and its shadow make a right angle. Because they are the same length, they form an isosceles right triangle (45/45/90). We can use the Pythagorean Theorem to find the hypotenuse. 
. In this case, 
. Therefore, we do 
. So 
.
The pole and its shadow make a right angle. Because they are the same length, they form an isosceles right triangle (45/45/90). We can use the Pythagorean Theorem to find the hypotenuse.
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Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
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Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above
Find the length of the hypotenuse.
Find the length of the hypotenuse.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of 
and 
and a hypotenuse of 
,
Now, plug in the values of 
and 
from the question.
Solve.
Simplify.
To find the hypotenuse of a triangle, you can use the Pythagorean Theorem. For any right triangle with leg lengths of
Now, plug in the values of
Solve.
Simplify.
Compare your answer with the correct one above