How to find the perimeter of a 45/45/90 right isosceles triangle - ACT Math
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A tree is 
feet tall and is planted in the center of a circular bed with a radius of 
feet. If you want to stabalize the tree with ropes going from its midpoint to the border of the bed, how long will each rope measure?
A tree is
This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem 
giving us 
. If 
then 
.
This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem
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What is the perimeter of an isosceles right triangle with an hypotenuse of length 
?
What is the perimeter of an isosceles right triangle with an hypotenuse of length
Your right triangle is a 
triangle. It thus looks like this:
Now, you know that you also have a reference triangle for 
triangles. This is:
This means that you can set up a ratio to find 
. It would be:
Your triangle thus could be drawn like this:
Now, notice that you can rationalize the denominator of 
:
Thus, the perimeter of your figure is:
Your right triangle is a
Now, you know that you also have a reference triangle for
This means that you can set up a ratio to find
Your triangle thus could be drawn like this:
Now, notice that you can rationalize the denominator of
Thus, the perimeter of your figure is:
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What is the perimeter of an isosceles right triangle with an area of 
?
What is the perimeter of an isosceles right triangle with an area of
Recall that an isosceles right triangle is also a 
triangle. Your reference figure for such a shape is:
or 
Now, you know that the area of a triangle is:
For this triangle, though, the base and height are the same. So it is:
Now, we have to be careful, given that our area contains 
. Let's use 
, for "side length":
Thus, 
. Now based on the reference figure above, you can easily see that your triangle is:
Therefore, your perimeter is:
Recall that an isosceles right triangle is also a
or
Now, you know that the area of a triangle is:
For this triangle, though, the base and height are the same. So it is:
Now, we have to be careful, given that our area contains
Thus,
Therefore, your perimeter is:
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An isosceles right triangle has a hypotenuse of length 
. What is the perimeter of this triangle, in terms of 
?
An isosceles right triangle has a hypotenuse of length
The ratio of sides to hypotenuse of an isosceles right triangle is always 
. With this in mind, setting 
as our hypotenuse means we must have leg lengths equal to:
Since the perimeter has two of these legs, we just need to multiply this by 
and add the result to our hypothesis:
So, our perimeter in terms of 
is:
The ratio of sides to hypotenuse of an isosceles right triangle is always
Since the perimeter has two of these legs, we just need to multiply this by
So, our perimeter in terms of
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