Question

1. what is the ratio of the sides for triangle xyc? image of a geometric figure with triangle and angles, and multiple choice options: 1:2:√3, 1:1:√2, 1:2:√2, 1:1:√3

Explanation:

Step1: Identify Triangle Type

Triangle XYC has a right angle at Y (since XY is perpendicular to YC, as indicated by the right angle symbol) and an angle of \(45^\circ\) at X. So, it's a right - isosceles triangle (because the sum of angles in a triangle is \(180^\circ\), so the third angle at C is also \(45^\circ\)). In a right - isosceles triangle, the legs are equal, and the hypotenuse \(h\) is related to the leg length \(l\) by \(h = l\sqrt{2}\) (from the Pythagorean theorem: \(h^{2}=l^{2}+l^{2}=2l^{2}\), so \(h = l\sqrt{2}\)).

Step2: Determine Side Ratios

Given that YC = 5 cm (from the diagram) and since it's a right - isosceles triangle, XY should also be 5 cm (because the two legs of a right - isosceles triangle are equal). Let the length of the legs be \(l = 5\) cm, then the hypotenuse \(XC=\sqrt{5^{2}+5^{2}}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}\) cm.

If we take the ratio of the sides (leg: leg: hypotenuse), we have \(5:5:5\sqrt{2}\). Dividing each term by 5, we get the ratio \(1:1:\sqrt{2}\).

Answer:

\(1:1:\sqrt{2}\) (corresponding to the option with this ratio, e.g., if the options are as given, the option with \(1:1:\sqrt{2}\))