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.
ANS:a=1.8cm,b=2.4cm
To find the lengths of sides a and b in the right triangle, given the conditions stated, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Let's set up the equation for each condition:
1. Sides a is twice as long as side b:
To represent this condition, we can write a = 2b.
Using the Pythagorean theorem:
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 = √(1/5)
b = 1/√5 (Exact length of side b)
b ≈ 0.44721 (Approximate length of side b)
Substitute the value of b back into the equation to find the length of side a:
a = 2b = 2(1/√5) = 2/√5 ≈ 0.89443 (Approximate length of side a)
2. Sides b is twice as long as side a:
To represent this condition, we can write b = 2a.
Using the Pythagorean theorem:
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 = √(1/5)
a = 1/√5 (Exact length of side a)
a ≈ 0.44721 (Approximate length of side a)
Substitute the value of a back into the equation to find the length of side b:
b = 2a = 2(1/√5) = 2/√5 ≈ 0.89443 (Approximate length of side b)
So, for each condition, side a is approximately 0.89443 units and side b is approximately 0.44721 units.