Question

a right - angled triangle abc with ab = 12 yd, ac = 15 yd. bc = ___

Explanation:

Step1: Apply Pythagorean theorem

In right - triangle \(ABC\) with right - angle at \(B\), the Pythagorean theorem states that \(AC^{2}=AB^{2}+BC^{2}\), where \(AC = 15\) yd and \(AB = 12\) yd. We want to find \(BC\), so we can rewrite the formula as \(BC=\sqrt{AC^{2}-AB^{2}}\).

Step2: Substitute the values

Substitute \(AC = 15\) and \(AB = 12\) into the formula: \(BC=\sqrt{15^{2}-12^{2}}=\sqrt{(15 + 12)(15 - 12)}\) (using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\)). Then \(BC=\sqrt{(27)(3)}=\sqrt{81}\).

Step3: Calculate the square root

\(\sqrt{81}=9\) yd.

Answer:

9 yd