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.
if b=2a, then
a^2 + (2a)^2 = 1
5a^2 = 1
a = 1/√5
similarly for a=2b
first case 1: a = 2b
so a^2 + b^2 = 1^2
(2b)^2 + b^2 = 1
5b^2 = 1
b^2 = 1/5
b = 1/√5
one side is 1/√5, the other is 2/√5
you do case 2
To find the lengths of sides a and b in the right triangle, given the conditions, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, side c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Let's solve each scenario step by step:
1. Sides a is twice as long as side b:
Let's assume b as the shorter side. Since a is twice as long, we can express it as a = 2b.
Using the Pythagorean theorem, we have:
c^2 = a^2 + b^2
(1^2) = ((2b)^2) + (b^2)
1 = 4b^2 + b^2
1 = 5b^2
b^2 = 1/5
b = sqrt(1/5)
Now, substituting the value of b in terms of a:
a = 2b
a = 2 * sqrt(1/5)
a = sqrt(4/5)
So, the lengths of sides a and b in this scenario are a = sqrt(4/5) and b = sqrt(1/5).
2. Sides b is twice as long as side a:
Let's assume a as the shorter side. Since b is twice as long, we can express it as b = 2a.
Using the Pythagorean theorem, we have:
c^2 = a^2 + b^2
(1^2) = (a^2) + ((2a)^2)
1 = a^2 + 4a^2
1 = 5a^2
a^2 = 1/5
a = sqrt(1/5)
Now, substituting the value of a in terms of b:
b = 2a
b = 2 * sqrt(1/5)
b = sqrt(4/5)
So, the lengths of sides a and b in this scenario are a = sqrt(1/5) and b = sqrt(4/5).
In both cases, the lengths of a and b are expressed in terms of square roots.