Question

find the value of x.
3√2
x
45° (right triangle with one leg 3√2, one angle 45° and hypotenuse x)

Explanation:

Step1: Identify Triangle Type

The triangle is a right - angled isosceles triangle (one angle is \(90^{\circ}\), another is \(45^{\circ}\), so the third angle is also \(45^{\circ}\)). In a \(45 - 45 - 90\) triangle, the hypotenuse \(c\) and the leg \(a\) are related by \(c=a\sqrt{2}\), or we can also use trigonometric ratios. Let's use trigonometry. We know that \(\sin(45^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to the \(45^{\circ}\) angle is \(3\sqrt{2}\), and the hypotenuse is \(x\).

Step2: Apply Sine Ratio

We know that \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\), and \(\sin(45^{\circ})=\frac{3\sqrt{2}}{x}\). So we can set up the equation \(\frac{\sqrt{2}}{2}=\frac{3\sqrt{2}}{x}\). Cross - multiply: \(x\times\sqrt{2}=2\times3\sqrt{2}\).

Step3: Solve for x

Divide both sides of the equation \(x\sqrt{2} = 6\sqrt{2}\) by \(\sqrt{2}\). We get \(x = 6\). Alternatively, since in a \(45 - 45 - 90\) triangle, if the leg length is \(a\), the hypotenuse \(h=a\sqrt{2}\) is not the way here (wait, no, in our case, the leg is \(3\sqrt{2}\), and we can also think that in a \(45 - 45 - 90\) triangle, \(\cos(45^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\), the adjacent side to \(45^{\circ}\) is equal to the other leg (since it's isosceles), and the hypotenuse is \(x\). But the first method using sine also works. Another way: in a \(45 - 45 - 90\) triangle, the hypotenuse is leg \(\times\sqrt{2}\)? Wait, no, actually, if the legs are of length \(l\), hypotenuse \(h = l\sqrt{2}\). But in our case, one leg is \(3\sqrt{2}\), and we can also use the formula for hypotenuse in terms of leg. Wait, let's re - examine. Let the legs be \(a\) and \(b\), hypotenuse \(c\). In \(45 - 45 - 90\), \(a = b\), and \(c=a\sqrt{2}\). Here, one leg \(a = 3\sqrt{2}\), so hypotenuse \(c=(3\sqrt{2})\times\sqrt{2}\)? Wait, no, that would be wrong. Wait, no, \(\sin(45^{\circ})=\frac{\text{opposite}}{x}=\frac{3\sqrt{2}}{x}\), and \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\), so \(\frac{3\sqrt{2}}{x}=\frac{\sqrt{2}}{2}\). Cross - multiplying: \(x\times\sqrt{2}=3\sqrt{2}\times2\), \(x\sqrt{2}=6\sqrt{2}\), divide both sides by \(\sqrt{2}\), \(x = 6\).

Answer:

\(x = 6\)