Question

find the value of $x$.
$15\sqrt{10}$
$x$
$45^{\circ}$

Explanation:

Step1: Identify triangle type

The triangle is right - angled with one angle \(45^{\circ}\), so it's a \(45 - 45-90\) triangle (isosceles right triangle), but wait, no—wait, the side given is one leg, and we can also use trigonometry. Let's use cosine or sine. Since it's a right triangle, \(\sin(45^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to \(45^{\circ}\) is \(15\sqrt{10}\), and the hypotenuse is \(x\).
\(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\), so \(\frac{\sqrt{2}}{2}=\frac{15\sqrt{10}}{x}\)

Step2: Solve for \(x\)

Cross - multiply: \(x\times\sqrt{2}=2\times15\sqrt{10}\)
\(x=\frac{30\sqrt{10}}{\sqrt{2}}\)
Simplify \(\frac{\sqrt{10}}{\sqrt{2}}=\sqrt{\frac{10}{2}}=\sqrt{5}\)
So \(x = 30\sqrt{5}\)
Wait, alternatively, since in a \(45 - 45-90\) triangle, legs are equal, but wait, no—wait, the angle is \(45^{\circ}\), so the two legs are equal? Wait, no, the right angle, one angle \(45^{\circ}\), so the other non - right angle is also \(45^{\circ}\), so legs are equal. Wait, but the side given is a leg, length \(15\sqrt{10}\), and hypotenuse \(x\). In a \(45 - 45-90\) triangle, hypotenuse \(= \text{leg}\times\sqrt{2}\). Wait, that's a mistake earlier. Let's correct:
In a \(45 - 45-90\) triangle, if leg length is \(a\), hypotenuse \(c=a\sqrt{2}\). Wait, but here, if the leg is \(15\sqrt{10}\), then hypotenuse \(x = 15\sqrt{10}\times\sqrt{2}=15\sqrt{20}=15\times2\sqrt{5}=30\sqrt{5}\). Wait, no, \(\sqrt{10}\times\sqrt{2}=\sqrt{20} = 2\sqrt{5}\), so \(15\times2\sqrt{5}=30\sqrt{5}\). Wait, but let's check with trigonometry again. \(\cos(45^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\), adjacent side is also a leg (since it's \(45^{\circ}\)), so adjacent side is equal to opposite side. Wait, no, the given side is one leg, so both legs are \(15\sqrt{10}\), and hypotenuse is \(x\). Then by Pythagoras: \(x^{2}=(15\sqrt{10})^{2}+(15\sqrt{10})^{2}\)
\(x^{2}=225\times10 + 225\times10=2250+2250 = 4500\)
\(x=\sqrt{4500}=\sqrt{900\times5}=30\sqrt{5}\)
Yes, that's correct.

Answer:

\(30\sqrt{5}\)