The length of a hypotenuse of a right triangle is 30 inches. Both legs of the right triangle are the same length. What are the lengths of the triangles legs?

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Answer 1 · 2018-02-13T00:21:36

See a solution process below:

We can use the Pythagorean Theorem to solve this problem.

The Pythagorean Theorem states, for right traingles:

#a^2 + b^2 = c^2#

Where:

#a# and #b# are the lengths of the legs of the right triangle.

#c# is the length of the hypotenuse.

In this problem we are told the two legs are the same length so we can write.

#a^2 + a^2 = c^2#

Or

#2a^2 = c^2#

We are also told the hypotenuse is 30 inches long. So, we can substitute for #c# and solve for #a# giving:

#2a^2 = (30"in")^2#

#2a^2 = 900"in"^2#

#(2a^2)/color(red)(2) = (900"in"^2)/color(red)(2)#

#a^2 = 450"in"^2#

#sqrt(a^2) = sqrt(450"in"^2)#

#a = sqrt(225"in"^2 * 2)#

#a = sqrt(225"in"^2)sqrt(2)#

#a = 15"in"sqrt(2)#

Or, in you need this as a number without a radical:

#a = 21.2"in"# rounded to the nearest tenth of an inch.

Answer 2 · 2018-02-13T02:47:00

Length of a leg #vec(AC) = vec(BC) = color(green)(21.21)#

enter image source here

In triangle ABC in the above figure' hypotenuse vec(AB) = 30"

#hatB, hatA# are equal and #hatC = 90^0#

#:. hatB = hatA = (180 - 90)/2 = 45^0#

Using trigonometric functions,

#vec(AC) = vec(AB) * sin B = 30 * sin(45) = 30 * (1/sqrt2) = 21.21"#

#vec(BC) = vec(AB) = 21.21"#