Question

solve for n. do not convert your answer to decimal form.
draw
12
show your work here
hint: to add the square root symbol (√□), type
oot\

Explanation:

Step1: Identify triangle type

It's a right - isosceles triangle (one angle \(45^\circ\), right angle), so legs are equal, and we can use trigonometry or Pythagorean theorem. Let's use cosine: \(\cos(45^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\cos(45^\circ)=\frac{12}{n}\)

Step2: Recall \(\cos(45^\circ)\) value

\(\cos(45^\circ)=\frac{\sqrt{2}}{2}\), so \(\frac{\sqrt{2}}{2}=\frac{12}{n}\)

Step3: Solve for \(n\)

Cross - multiply: \(n\sqrt{2}=24\), then \(n = \frac{24}{\sqrt{2}}\)
Rationalize the denominator: \(n=\frac{24\sqrt{2}}{2}=12\sqrt{2}\) (or use Pythagorean theorem: legs are 12 and 12, so \(n=\sqrt{12^{2}+12^{2}}=\sqrt{144 + 144}=\sqrt{288}=12\sqrt{2}\))

Answer:

\(12\sqrt{2}\)