Question

in the following figure. find the length of the unknown side of the triangle. give an exact answer then an answer correct to 5 decimal places. to enter √5 type in sqrt(5)

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 4\), \(b = 9\) and hypotenuse \(x\), the Pythagorean theorem is \(a^{2}+b^{2}=x^{2}\). So \(x=\sqrt{4^{2}+9^{2}}\).

Step2: Calculate the squares

\(4^{2}=16\) and \(9^{2}=81\). Then \(x = \sqrt{16 + 81}\).

Step3: Add the values inside the square - root

\(16+81 = 97\), so \(x=\sqrt{97}\approx9.84886\). But if we assume the legs are \(a = 4\) and \(b\) is the unknown side and hypotenuse \(x = 9\), then by the Pythagorean theorem \(b=\sqrt{x^{2}-a^{2}}=\sqrt{9^{2}-4^{2}}=\sqrt{81 - 16}=\sqrt{65}\approx 8.06226\). If we assume the problem is asking for the hypotenuse of a right - triangle with legs 4 and 9, then:
\(x=\sqrt{4^{2}+9^{2}}=\sqrt{16 + 81}=\sqrt{97}\approx9.84886\). If we assume the problem is set up differently and we are solving for a non - hypotenuse side (which seems less likely given the way the problem is set up visually), say the vertical side is the unknown and hypotenuse is 9 and base is 4, then \(x=\sqrt{9^{2}-4^{2}}=\sqrt{81-16}=\sqrt{65}\approx8.06226\). But if we go by the most common interpretation of finding the hypotenuse of a right - triangle with legs 4 and 9:
\(x=\sqrt{4^{2}+9^{2}}=\sqrt{16 + 81}=\sqrt{97}\)

Answer:

\(x=\sqrt{97}\approx9.84886\)