Question: Find the size of angle $XYZ$. Give your answer to 1 decimal place. $Z$ $12$ cm $X$ $4$ cm $Y$ Right triangle $XYZ$ with a right angle at $X$; $ZX$ is the vertical side labeled $12$ cm, $XY$ is the horizontal side labeled $4$ cm, and the shaded angle is at $Y$. position_hint = center

1. **Problem Statement:** We need to find the size of angle $XYZ$ in a right triangle $XYZ$ where the right angle is at $X$. The side $ZX$ (vertical) is $12$ cm and side $XY$ (horizontal) is $4$ cm. 2. **Identify the angle and sides:** Angle $XYZ$ is at vertex $Y$. The sides adjacent to angle $Y$ are $XY$ (horizontal) and $YZ$ (hypotenuse). We know $XY=4$ cm and $ZX=12$ cm. 3. **Use trigonometric ratios:** Since angle $X$ is right, triangle $XYZ$ is right-angled. To find angle $Y$, we use the tangent function which relates the opposite side to the adjacent side: $$\tan(\angle XYZ) = \frac{\text{opposite side}}{\text{adjacent side}}$$ Here, opposite side to angle $Y$ is $ZX=12$ cm and adjacent side is $XY=4$ cm. 4. **Calculate tangent:** $$\tan(\angle XYZ) = \frac{12}{4} = 3$$ 5. **Find the angle:** Use the inverse tangent (arctan) to find the angle: $$\angle XYZ = \tan^{-1}(3)$$ 6. **Evaluate the angle:** Using a calculator, $$\angle XYZ \approx 71.565\degree$$ 7. **Round to 1 decimal place:** $$\angle XYZ \approx 71.6\degree$$ **Final answer:** The size of angle $XYZ$ is approximately $71.6\degree$.