Question

find v.
7\sqrt{2} km
45°
45°
v
write your answer in simplest radical form.
kilometers

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) right - triangle. In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the sides are in the ratio \(a:a:a\sqrt{2}\), where \(a\) is the length of the legs and \(a\sqrt{2}\) is the length of the hypotenuse.

Step2: Apply the ratio formula

Let the length of each leg be \(x\). The hypotenuse \(c = 7\sqrt{2}\) km. Using the formula \(c=a\sqrt{2}\) (for \(45 - 45-90\) triangle), if \(c = 7\sqrt{2}\), then \(a\sqrt{2}=7\sqrt{2}\). Solving for \(a\) (one of the legs), we divide both sides by \(\sqrt{2}\), getting \(a = 7\) km. But we can also use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Since \(a = b\) (because the two non - hypotenuse angles are equal), we have \(2a^{2}=c^{2}\). Substitute \(c = 7\sqrt{2}\), then \(2a^{2}=(7\sqrt{2})^{2}=98\). So \(a^{2}=49\), and \(a = 7\). Then, using the formula for the hypotenuse \(v=\sqrt{a^{2}+(7\sqrt{2})^{2}}\) (wait, no, wrong approach. Let's start over with correct ratio).

In a \(45-45 - 90\) triangle, if the legs are of length \(l\) and the hypotenuse is \(h\), \(h = l\sqrt{2}\). Here, we know one leg \(l_1=7\) (from \(7\sqrt{2}\div\sqrt{2}=7\)) and we want to find the hypotenuse \(v\). Wait, no, wait the given side \(7\sqrt{2}\) is a leg? No, no. Wait the right - angled side (the side adjacent to the two \(45^{\circ}\) angles) is \(7\sqrt{2}\). Wait no, the triangle has a right angle (\(90^{\circ}\)) and two \(45^{\circ}\) angles. The side given as \(7\sqrt{2}\) is a leg. Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), and \(a = b = 7\sqrt{2}\) (no, wait no. Wait if the two non - hypotenuse angles are \(45^{\circ}\), then the legs are equal. Let the legs be \(x\). Then \(x^{2}+x^{2}=v^{2}\). Given \(x = 7\sqrt{2}\) (no, wait no. Wait the side \(7\sqrt{2}\) is a leg. Wait no, in a \(45-45 - 90\) triangle, hypotenuse \(h\), legs \(l\). \(h=l\sqrt{2}\). If \(l = 7\sqrt{2}\), then \(h=(7\sqrt{2})\times\sqrt{2}=14\)

Answer:

\(14\) kilometers