Question

what is the length of side a? image of a right triangle with legs a and 3, hypotenuse 4

Explanation:

Step1: Identify triangle type

It's 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 \(b\), \(a^{2}+b^{2}=c^{2}\). Here, the hypotenuse \(c = 4\) and one leg \(b = 3\), and we want to find \(a\).

Step2: Rearrange the formula

We can rearrange the Pythagorean theorem to solve for \(a\): \(a=\sqrt{c^{2}-b^{2}}\).

Step3: Substitute the values

Substitute \(c = 4\) and \(b = 3\) into the formula: \(a=\sqrt{4^{2}-3^{2}}=\sqrt{16 - 9}=\sqrt{7}\approx2.65\) (if we want a decimal approximation) or leave it as \(\sqrt{7}\).

Answer:

The length of side \(a\) is \(\sqrt{7}\) (or approximately \(2.65\)).