The radius of this circle is one unit.
Find the exact lengths of the legs in the right triangle if:
Sides a is twice as long as side b.
And Sides b is twice as long as side a.
Basically the hypotenuse of this triangle is 1 and we are trying to use the pythagorean theorem to find the lengths of side a and b according to the terms described above.
Pythagorean must always be capitalized as it is a proper noun.
What is the equation for the Pythagorean theorem, also know as the Pythagorean equation?
Once you have this equation, you should be able to solve this problem.
To find the lengths of sides a and b in the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's consider the case where side a is twice as long as side b.
We are given that the radius of the circle (which is also the length of the hypotenuse) is 1 unit. So, the hypotenuse of the right triangle is 1 unit.
We'll denote side a as 2b (twice the length of side b), so side b will be b units.
According to the Pythagorean theorem, we have:
(2b)^2 + b^2 = 1^2
Expanding and simplifying the above equation:
4b^2 + b^2 = 1
Combining like terms:
5b^2 = 1
Now, we can solve for b by isolating it:
Dividing both sides by 5, we get:
b^2 = 1/5
Taking the square root of both sides:
b = √(1/5) or b = -√(1/5)
Since we are dealing with lengths, we'll take the positive value:
b = √(1/5)
Now that we have the value of b, we can find the length of side a, which is twice the length of side b:
a = 2b = 2√(1/5) = 2/√5 * (√5/√5) = 2√5/5
So, the lengths of sides a and b in the right triangle are a = 2√5/5 and b = √(1/5), respectively, when side a is twice as long as side b.
To solve the second case, where side b is twice as long as side a, you would follow a similar process but interchange the roles of a and b in the equation. The final solution would be a = √(1/5) and b = 2√5/5.