Question

q.5 special right triangles
find n.
right triangle with a right angle, two 45° angles, one leg labeled n, the other leg labeled \\(\sqrt{6}\\) mm
write your answer in simplest radical form.
blank millimeters

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 triangle (right - isosceles triangle), where the legs are equal.

Step2: Simplify \(\sqrt{6}\)? Wait, no. Wait, in a 45 - 45 - 90 triangle, the legs are equal. Wait, the given leg is \(\sqrt{6}\)? Wait, no, wait the triangle has two 45 - degree angles and a right angle, so it's an isosceles right triangle, so the two legs are equal. Wait, but maybe I misread. Wait, the side labeled \(\sqrt{6}\) is a leg, and \(n\) is also a leg? Wait, no, wait maybe the hypotenuse? Wait, no, the right angle is between \(n\) and the other leg. Wait, no, let's re - examine. The triangle has angles 45°, 45°, and 90°, so it's an isosceles right triangle. In an isosceles right triangle, the legs are equal. Wait, but maybe the side with length \(\sqrt{6}\) is a leg, and \(n\) is also a leg? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, no, in a 45 - 45 - 90 triangle, the ratio of legs to hypotenuse is \(1:1:\sqrt{2}\). Wait, maybe the given side is the hypotenuse? Wait, no, the right angle is at the corner with the red square, so the two legs are \(n\) and the side with \(\sqrt{6}\), and the hypotenuse is opposite the right angle. Wait, but the angles are 45°, 45°, 90°, so the legs are equal. Wait, so if one leg is \(\sqrt{6}\), then the other leg \(n\) is also \(\sqrt{6}\)? Wait, no, that seems too easy. Wait, no, maybe the given side is the hypotenuse. Wait, let's check the angle labels. The two non - right angles are 45°, so the triangle is isosceles with legs equal. So if one leg is \(\sqrt{6}\), then \(n=\sqrt{6}\)? Wait, no, that can't be. Wait, maybe I misread the length. Wait, the side is labeled \(\sqrt{6}\) mm? Wait, no, maybe it's a different length. Wait, no, the problem says "Find \(n\)". Let's start over.

In a 45 - 45 - 90 triangle, the legs are congruent (equal in length) because the angles opposite them (the 45° angles) are equal. The right angle is between the two legs. So if one leg has length \(a\) and the other leg has length \(b\), then \(a = b\) in a 45 - 45 - 90 triangle.

Looking at the triangle, one leg is \(n\) and the other leg is \(\sqrt{6}\) mm (since the two non - right angles are 45° each). Therefore, by the property of 45 - 45 - 90 triangles (isosceles right triangles), the two legs are equal.

Wait, but maybe I made a mistake. Wait, no, in a triangle with angles 45°, 45°, and 90°, the sides opposite the 45° angles (the legs) are equal. So if one leg is \(\sqrt{6}\), then the other leg \(n\) is also \(\sqrt{6}\). But let's verify. The length of \(\sqrt{6}\) is already in simplest radical form.

Wait, but maybe the given side is the hypotenuse. Let's assume that the side with length \(\sqrt{6}\) is the hypotenuse. Then, in a 45 - 45 - 90 triangle, the length of each leg \(l\) is given by \(l=\frac{hypotenuse}{\sqrt{2}}\). So if hypotenuse \(h = \sqrt{6}\), then \(l=\frac{\sqrt{6}}{\sqrt{2}}=\sqrt{3}\). But that contradicts the earlier thought. Wait, which is it?

Wait, the right angle is at the corner with the red square. So the two sides forming the right angle are the legs (\(n\) and the side with \(\sqrt{6}\)), and the side opposite the right angle is the hypotenuse. The angles opposite the legs are 45° each, so the legs are equal. Therefore, \(n=\sqrt{6}\). But that seems too simple. Wait, maybe the side labeled \(\sqrt{6}\) is a typo, but according to the problem, we have to go with what's given.

Wait, no, maybe I misread the side length. Wait, the side is labeled \(\sqrt{6}\) mm? Wait, no, maybe it's a different length. Wait, no, the problem says "Write your answer in simplest radical form". So if the leg is \(\sqrt{6}\), then \(n = \sqrt{6}\). But let's check the ratio again. In a 45 - 45 - 90 triangle, legs are equal. So if one leg is \(x\), the other leg is \(x\), and hypotenuse is \(x\sqrt{2}\). So if the given side is a leg, then \(n\) (the other leg) is equal to it. So if the given leg is \(\sqrt{6}\), then \(n=\sqrt{6}\).

Answer:

\(\sqrt{6}\)