Question 696179: Find the hypotenuse of each isosceles right triangle when the legs are of the given measure.
Given = 3 sqrt 2
Found 2 solutions by Alan3354, RedemptiveMath:
Answer by Alan3354(69443)   (Show Source):
Answer by RedemptiveMath(80)  (Show Source):
You can put this solution on YOUR website! We are given this triangle to be right and to be isosceles. This lets us know two things: 1) We can use the properties of a right triangle, and 2) we can use the properties of an isosceles triangle. Some of these properties will help us figure out the answer to the problem. Since we are dealing with side lengths, it will be helpful to keep the Pythagorean theorem in mind. We also know that the legs of an isosceles triangle are congruent. With these two pieces of information we can solve the problem. Let us first recognize the isosceles triangle property.
We have two congruent legs to an isosceles triangle. Since we already know that one of those legs equals 3√2, the other must be the same length. We still need to find what the hypotenuse measures, and we can do this by examining (and using) the Pythagorean theorem.
The Pythagorean theorem lets us know that a right triangle is arranged in such a way that its sides' measures can be written as so:
a^2 + b^2 = c^2,
where a and b are the legs and c is the hypotenuse of the triangle. We know that the hypotenuse designates the longest side. The legs are the other two sides. Since we know two of this triangle's sides, we can find c by using algebra:
a^2 + b^2 = c^2
(3√2)^2 + (3√2)^2 = c^2 (plug 3√2 into a and b)
(3^2)(√2^2) + (3^2)(√2^2) = c^2 (distribute the exponent to both items in the parentheses; we must remember that 3 is being multiplied by a separate item √2).
(3√2)(3√2) + (3√2)(3√2) = c^2 (another way we can write the above equation)
9(2) + 9(2) = c^2 (simplify the multiplication; remember that a square root and a squared operation cancel out)
18 + 18 = c^2 (multiplication)
36 = c^2 (combine like terms)
c^2 = 36 (you don't have to do this step; this just puts the variable first)
c = √36 (square root both sides to get c by itself)
c = √36 = 6 or -6 (simplify √36).
We must remember that any positive number has a positive and negative square root. However, the principal root (positive) is the one we usually are concerned with when we deal with measurement. We cannot have a negative length in simple ideology, so we only look at the positive root in these instances. This is not be the case when finding roots to quadratic functions, bank account balances, complex numbers, and so forth. Therefore, c = 6.
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