Question

gillian is a farmer who is building a pen for some animals. the first side of the pen is the longest, measuring 27 feet. the second side of the pen is 15 feet long. how long is the third side of the pen? round your answer to two decimal places.

(image of a right triangle with one leg 15 ft, hypotenuse 27 ft, and the right angle at the intersection of the 15 ft leg and the third side)

show your work here

hint: to add the square root symbol (√) type \sqrt\

Explanation:

Step1: Identify the triangle type

The pen is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with hypotenuse \(c\) and legs \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}\). Here, the hypotenuse \(c = 27\) feet and one leg \(a = 15\) feet. We need to find the other leg \(b\). Rearranging the formula for \(b\), we get \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values

Substitute \(c = 27\) and \(a = 15\) into the formula: \(b=\sqrt{27^{2}-15^{2}}\). First, calculate \(27^{2}=729\) and \(15^{2}=225\). Then, \(27^{2}-15^{2}=729 - 225=504\). So, \(b=\sqrt{504}\).

Step3: Simplify and round

Simplify \(\sqrt{504}\). We know that \(504 = 36\times14\), so \(\sqrt{504}=\sqrt{36\times14}=6\sqrt{14}\approx6\times3.7417\approx22.45\) (rounded to two decimal places).

Answer:

\(22.45\) feet