Question

1. what is the length of segment xy?

options: (5sqrt{2}mathrm{cm}), (10mathrm{cm}), (5mathrm{cm}), (10sqrt{2}mathrm{cm}) (and a diagram of a triangle with points a, b, c, x, y, angles 30°, 45°, side bc = 10 cm, yc = 5 cm)

Explanation:

Step1: Analyze Triangle XYC

Triangle XYC is a right - angled triangle (since $\angle XYC = 90^{\circ}$) with $\angle XCY=45^{\circ}$. In a right - angled triangle, if one of the non - right angles is $45^{\circ}$, then the triangle is an isosceles right - angled triangle, which means the legs are equal. We know that $YC = 5$ cm. Let the length of $XY$ be $x$ and the length of $YC$ be $y$. In $\triangle XYC$, $\tan(45^{\circ})=\frac{XY}{YC}$. Since $\tan(45^{\circ}) = 1$, we have $\frac{XY}{YC}=1$.

Step2: Calculate XY

We know that $YC = 5$ cm. From the property of the isosceles right - angled triangle (or from the tangent formula), since $\tan(45^{\circ})=\frac{XY}{YC}$ and $\tan(45^{\circ}) = 1$, we can say that $XY=YC$. But wait, there is a mistake here. Wait, actually, looking at the triangle, $YC = 5$ cm, and $\angle XCY = 45^{\circ}$, $\angle XYC=90^{\circ}$, so triangle XYC is isosceles right - angled, so $XY = YC$? No, wait, no. Wait, $YC$ is 5 cm, and if we consider the other way, maybe we made a mistake. Wait, no, let's re - examine. Wait, the length of $BC$ is 10 cm, and $YC$ is 5 cm. Wait, in triangle XYC, $\angle XCY = 45^{\circ}$, $\angle XYC = 90^{\circ}$, so $XY=YC$? No, that can't be. Wait, no, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. In $\triangle XYC$, $\theta = 45^{\circ}$, opposite side to $45^{\circ}$ is $XY$, adjacent side is $YC$. So $\tan(45^{\circ})=\frac{XY}{YC}\Rightarrow1=\frac{XY}{5}\Rightarrow XY = 5$? No, that's not matching the options. Wait, maybe we misread the triangle. Wait, the length of $YC$ is 5 cm? Wait, no, maybe the length of $YC$ is 5 cm, but let's check the other triangle. Wait, no, maybe the triangle is such that $YC = 5$ cm, and $\angle XCY=45^{\circ}$, so $XY = YC$? But the options have $5\sqrt{2}$, 10, 5, etc. Wait, no, maybe I made a mistake in identifying the triangle. Wait, actually, in triangle XYC, it's a right - angled triangle with $\angle XCY = 45^{\circ}$, so it's an isosceles right - angled triangle, so $XY=YC$? But $YC = 5$ cm? No, that's not right. Wait, maybe the length of $YC$ is 5 cm, but the triangle is such that $XY$ and $YC$ are the legs. Wait, no, let's use the Pythagorean theorem. In an isosceles right - angled triangle, if the legs are of length $a$, then the hypotenuse is $a\sqrt{2}$. But here, if $\angle XCY = 45^{\circ}$, $\angle XYC = 90^{\circ}$, then $XY = YC$? Wait, no, $\angle XCY=45^{\circ}$, so the two legs $XY$ and $YC$ are equal. Wait, but if $YC = 5$ cm, then $XY = 5$ cm? But that's one of the options. Wait, but let's check again. Wait, the diagram shows that $YC = 5$ cm, and $\angle XCY = 45^{\circ}$, $\angle XYC=90^{\circ}$, so triangle XYC is isosceles right - angled, so $XY = YC = 5$ cm? But wait, the options have 5 cm as an option. Wait, but maybe I misread the triangle. Wait, no, let's see:

Wait, the right angle is at Y, so $\triangle XYC$ has right angle at Y, angle at C is $45^{\circ}$, so angle at X is also $45^{\circ}$, so $XY = YC$. Since $YC = 5$ cm, then $XY = 5$ cm? But the options have 5 cm as an option. Wait, but let's check the other parts. Wait, the length of BC is 10 cm, and YC is 5 cm, so BY is also 5 cm? Wait, maybe. So in $\triangle XYC$, right - angled at Y, angle at C is $45^{\circ}$, so $XY = YC = 5$ cm.

Step3: Verify with Options

Looking at the options, 5 cm is one of the options. So the length of segment XY is 5 cm.

Answer:

5 cm (the option corresponding to 5 cm)