Question

what is the length of \\(\overline{ab}\\)? round to the nearest tenth.
(there is a right triangle abc with right angle at c, ac = 10m, angle at a is 75°, and ab is labeled as x. there are four rectangular boxes with values 38.6m, 9.7m, 10.4m, 37.3m respectively.)

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\), \(\angle C = 90^\circ\), \(\angle A = 75^\circ\), and \(AC = 10\) m. We need to find the length of \(AB\) (hypotenuse). We can use the cosine function, which is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\cos(75^\circ)=\frac{AC}{AB}\), so \(AB=\frac{AC}{\cos(75^\circ)}\).

Step2: Calculate the value

First, find \(\cos(75^\circ)\). Using a calculator, \(\cos(75^\circ)\approx0.2588\). Then, substitute \(AC = 10\) m into the formula: \(AB=\frac{10}{\cos(75^\circ)}\approx\frac{10}{0.2588}\approx38.6\) m. Wait, no, wait. Wait, actually, in the right triangle, angle at \(A\) is \(75^\circ\), side \(AC\) is adjacent to angle \(A\), and \(AB\) is the hypotenuse. Wait, no, wait, maybe I mixed up. Wait, \(\cos(\angle A)=\frac{AC}{AB}\), so \(AB = \frac{AC}{\cos(\angle A)}\). But let's check again. Wait, angle at \(A\) is \(75^\circ\), \(AC\) is adjacent, \(AB\) is hypotenuse. So \(\cos(75^\circ)=\frac{AC}{AB}\), so \(AB=\frac{AC}{\cos(75^\circ)}\). But wait, maybe it's \(\cos(75^\circ)=\frac{AC}{AB}\), so \(AB=\frac{AC}{\cos(75^\circ)}\). Let's calculate \(\cos(75^\circ)\). \(\cos(75^\circ)=\cos(45^\circ + 30^\circ)=\cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.2588\). Then \(AB=\frac{10}{0.2588}\approx38.6\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, maybe it's the cosine of angle \(A\) is adjacent over hypotenuse. Wait, \(AC\) is adjacent to angle \(A\), \(AB\) is hypotenuse. So \(\cos(75^\circ)=\frac{AC}{AB}\), so \(AB=\frac{AC}{\cos(75^\circ)}\). But let's check with calculator. \(\cos(75^\circ)\approx0.2588\), so \(10\div0.2588\approx38.6\). But wait, the options have 38.6 m. Wait, but let's check again. Wait, maybe it's the other way. Wait, angle at \(A\) is \(75^\circ\), so the side opposite to angle \(A\) is \(BC\), adjacent is \(AC\), hypotenuse is \(AB\). So \(\cos(75^\circ)=\frac{AC}{AB}\), so \(AB=\frac{AC}{\cos(75^\circ)}\). So \(10\div\cos(75^\circ)\approx10\div0.2588\approx38.6\) m. So the length of \(AB\) is approximately 38.6 m.

Answer:

38.6 m