Question

find the value of x.
13√2
x
45°

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (one angle \(45^\circ\), right angle \(90^\circ\), so the third angle is also \(45^\circ\)). In a \(45 - 45 - 90\) triangle, the hypotenuse \(c\) and leg \(a\) are related by \(c=a\sqrt{2}\), or leg \(a=\frac{c}{\sqrt{2}}\). Here, the leg length is \(13\sqrt{2}\), and we need to find the hypotenuse \(x\).

Step2: Apply \(45 - 45 - 90\) Triangle Ratio

For a \(45 - 45 - 90\) triangle, the hypotenuse \(x\) is given by \(x=\text{leg}\times\sqrt{2}\)? Wait, no. Wait, in a \(45 - 45 - 90\) triangle, if the leg length is \(l\), the hypotenuse \(h = l\sqrt{2}\). Wait, no, actually, if the leg is \(l\), then hypotenuse \(h=l\sqrt{2}\)? Wait, no, let's recall: in a right - isosceles triangle (angles \(45^\circ,45^\circ,90^\circ\)), the ratio of leg : leg : hypotenuse is \(1:1:\sqrt{2}\). So if one leg is \(l\), the hypotenuse is \(l\sqrt{2}\), and if the hypotenuse is \(h\), then each leg is \(\frac{h}{\sqrt{2}}\).

In our triangle, the leg (the side with length \(13\sqrt{2}\)) and we need to find the hypotenuse \(x\). Since it's a \(45 - 45 - 90\) triangle, hypotenuse \(x=\text{leg}\times\sqrt{2}\)? Wait, no, wait. Wait, the leg is \(13\sqrt{2}\), so according to the ratio \(1:1:\sqrt{2}\), if the leg is \(l = 13\sqrt{2}\), then hypotenuse \(x=l\sqrt{2}\)? Wait, no, that would be wrong. Wait, let's do it with trigonometry. Let's use \(\sin(45^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to the \(45^\circ\) angle is \(13\sqrt{2}\), and the hypotenuse is \(x\). So \(\sin(45^\circ)=\frac{13\sqrt{2}}{x}\). We know that \(\sin(45^\circ)=\frac{\sqrt{2}}{2}\). So \(\frac{\sqrt{2}}{2}=\frac{13\sqrt{2}}{x}\). Cross - multiply: \(x\times\sqrt{2}=2\times13\sqrt{2}\). Divide both sides by \(\sqrt{2}\): \(x = 26\).

Alternatively, since it's a \(45 - 45 - 90\) triangle, the two legs are equal. So the other leg is also \(13\sqrt{2}\). Then by Pythagoras' theorem, \(x^{2}=(13\sqrt{2})^{2}+(13\sqrt{2})^{2}\). Calculate \((13\sqrt{2})^{2}=13^{2}\times2 = 169\times2=338\). So \(x^{2}=338 + 338=676\). Then \(x=\sqrt{676}=26\).

Answer:

\(26\)