Question

problem 3
calculate the exact value of the unknown side length in the right triangle.

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle (as there is a right - angle symbol). So we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if \(c\) is the hypotenuse) or \(a^{2}=c^{2}-b^{2}\) (if \(a\) is a leg and \(c\) is the hypotenuse and \(b\) is the other leg). Let the unknown side be \(x\), the hypotenuse be \(\sqrt{80}\) and one leg be \(\sqrt{68}\), and the other leg be \(x\).

Step2: Apply the Pythagorean theorem

Using the formula \(a^{2}=c^{2}-b^{2}\) (where \(c = \sqrt{80}\), \(b=\sqrt{68}\) and \(a=x\)):
\[

$$\begin{align*} x^{2}&=(\sqrt{80})^{2}-(\sqrt{68})^{2}\\ x^{2}&=80 - 68\\ x^{2}&=12 \end{align*}$$

\]

Step3: Solve for \(x\)

Take the square root of both sides. Since \(x\) represents a length, we take the positive square root. So \(x=\sqrt{12}=\sqrt{4\times3} = 2\sqrt{3}\)

Answer:

\(2\sqrt{3}\)