Question

question 44 of 50
the value of x is:
triangle with right angle, 60° angle, hypotenuse 7, side x
options: 7√2, 14, 3.5, 7, 7√3

Explanation:

Step1: Identify Triangle Type

The triangle is right - angled (one right angle) and has a \(60^{\circ}\) angle. So, the third angle is \(180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}\). In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side, the side opposite \(60^{\circ}\) is \(\sqrt{3}\) times the shortest side, and the hypotenuse (opposite \(90^{\circ}\)) is twice the shortest side. Here, the side of length \(7\) is opposite the \(30^{\circ}\) angle? Wait, no. Wait, the side labeled \(x\) is adjacent to the \(60^{\circ}\) angle, and the hypotenuse? Wait, no, let's re - examine. The right angle, \(60^{\circ}\) angle, so the side with length \(7\) is the hypotenuse? Wait, no. Wait, in a right - triangle, the sum of angles is \(180^{\circ}\). So angles are \(90^{\circ}\), \(60^{\circ}\), and \(30^{\circ}\). Let's denote the sides: let the side opposite \(30^{\circ}\) be \(a\), opposite \(60^{\circ}\) be \(a\sqrt{3}\), and hypotenuse be \(2a\). Now, the side of length \(7\) is the hypotenuse? Wait, no, the side labeled \(x\) is adjacent to the \(60^{\circ}\) angle, and the side of length \(7\) is the hypotenuse? Wait, no, let's look at the triangle again. The side with length \(7\) is the side opposite the \(60^{\circ}\) angle? No, wait, the angle of \(60^{\circ}\) has the side \(x\) as one of its adjacent sides, and the side of length \(7\) as the hypotenuse? Wait, no, in a right - triangle, \(\cos(60^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\). Wait, \(\cos(60^{\circ})=\frac{x}{7}\)? No, wait, \(\cos(60^{\circ}) = 0.5=\frac{x}{7}\)? No, that would give \(x = 3.5\), but wait, maybe I got the angle wrong. Wait, the angle of \(60^{\circ}\), the side opposite to \(30^{\circ}\) is the shortest side. Wait, let's use trigonometric ratios. Let's say the right angle is at the top, the \(60^{\circ}\) angle is at the left, so the side \(x\) is the side adjacent to the \(60^{\circ}\) angle, and the side of length \(7\) is the hypotenuse? No, wait, \(\cos(60^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(60^{\circ})=\frac{x}{7}\), then \(x = 7\times\cos(60^{\circ})=7\times0.5 = 3.5\). Wait, but let's check with the \(30 - 60 - 90\) triangle ratios. In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). If the hypotenuse is \(7\), then the side opposite \(30^{\circ}\) is \(\frac{7}{2}=3.5\), and the side opposite \(60^{\circ}\) is \(\frac{7\sqrt{3}}{2}\). Wait, no, maybe the side of length \(7\) is the side opposite the \(60^{\circ}\) angle. Then, \(\sin(60^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\), but no, let's use \(\cos(60^{\circ})=\frac{x}{7}\) is wrong. Wait, let's use \(\sin(30^{\circ})=\frac{x}{7}\), because \(\sin(30^{\circ}) = 0.5=\frac{x}{7}\), so \(x = 3.5\). Yes, that makes sense. Because the angle of \(30^{\circ}\) (since \(90 + 60+30 = 180\)) has the side \(x\) as its opposite side, and the hypotenuse is \(7\). So \(\sin(30^{\circ})=\frac{x}{7}\), and since \(\sin(30^{\circ})=\frac{1}{2}\), then \(x = 7\times\frac{1}{2}=3.5\).

Step2: Verify with Trigonometric Ratio

We know that in a right - triangle, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\) (since the other non - right angle is \(30^{\circ}\) as \(180 - 90 - 60=30\)), the opposite side to \(30^{\circ}\) is \(x\), and the hypotenuse is \(7\). So \(\sin(30^{\circ})=\frac{x}{7}\). Since \(\sin(30^{\circ})=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{x}{7}\). Cross - multiplying gives \(x=\frac{7}…

Answer:

\(3.5\) (or the option with \(3.5\))