1. **State the problem:** We are given a right triangle with vertices on the coordinate axes and need to find the range of the hypotenuse length based on the given points. 2. **Identify the points:** The base of the triangle lies on the x-axis from 0 to 5, so the base length is $5$ units. 3. **Vertical side values:** The vertical side has points labeled 12, 13, and 17, which represent possible heights of the triangle. 4. **Calculate the hypotenuse length:** Using the Pythagorean theorem, the hypotenuse $h$ is given by $$h = \sqrt{\text{base}^2 + \text{height}^2} = \sqrt{5^2 + y^2} = \sqrt{25 + y^2}$$ where $y$ is the vertical side length. 5. **Calculate hypotenuse for each height:** - For $y=12$: $$h = \sqrt{25 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ - For $y=13$: $$h = \sqrt{25 + 13^2} = \sqrt{25 + 169} = \sqrt{194} \approx 13.93$$ - For $y=17$: $$h = \sqrt{25 + 17^2} = \sqrt{25 + 289} = \sqrt{314} \approx 17.72$$ 6. **Determine the range:** The hypotenuse length ranges from the smallest value 13 to the largest value approximately 17.72. **Final answer:** The range of the hypotenuse length is $$[13, 17.72]$$.