QuestionFind the length of side in simplest radical form with a rational denominator. Answer Attempt 1 out of 2
Studdy Solution
STEP 1
What is this asking? We need to find the length of one leg of a 45-45-90 triangle, given that its hypotenuse is 9. Watch out! Don't forget to simplify the radical and rationalize the denominator!
STEP 2
1. Recall the 45-45-90 triangle relationships.
2. Set up an equation and solve for .
STEP 3
In a 45-45-90 triangle, the two legs are **congruent**, meaning they have the **same length**.
Let's call this length .
STEP 4
The **hypotenuse** is times the length of each **leg**.
So, if the leg has length , the hypotenuse has length .
STEP 5
We're given that the **hypotenuse** is **9**.
We also know from the 45-45-90 triangle relationships that the hypotenuse is .
So, we can set up the equation: .
STEP 6
Now, let's **isolate** by dividing both sides of the equation by .
Remember, we're dividing by because that will turn the on the left into a **1**, and is just !
STEP 7
We need to **rationalize the denominator**, which means getting rid of the square root in the bottom.
We can do this by multiplying both the **numerator** and the **denominator** by .
Why both?
Because is just **1**, and multiplying by 1 doesn't change the value of our fraction!
Now our answer has a **rational denominator**!
STEP 8
The length of side is .
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