1. **Problem Statement:** We are given a right triangle with a vertical leg of length 3.2 cm, a horizontal leg of length 1.7 cm, and a slanted side (hypotenuse) that forms an angle of approximately 22° with the horizontal. We need to find the length of the slanted side. 2. **Formula Used:** In a right triangle, the Pythagorean theorem relates the sides: $$c = \sqrt{a^2 + b^2}$$ where $c$ is the hypotenuse, and $a$, $b$ are the legs. 3. **Step-by-step Solution:** - Given vertical leg $a = 3.2$ cm and horizontal leg $b = 1.7$ cm. - Calculate the hypotenuse: $$c = \sqrt{3.2^2 + 1.7^2}$$ $$= \sqrt{10.24 + 2.89}$$ $$= \sqrt{13.13}$$ $$\approx 3.625$$ cm 4. **Verification with angle:** The angle near 22° suggests the slanted side is opposite that angle. Using sine: $$\sin(22^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3.2}{c}$$ Rearranged: $$c = \frac{3.2}{\sin(22^\circ)}$$ Calculate: $$\sin(22^\circ) \approx 0.3746$$ $$c \approx \frac{3.2}{0.3746} \approx 8.54$$ cm This conflicts with the Pythagorean calculation, indicating the 22° angle is likely between the hypotenuse and the horizontal leg, not opposite the vertical leg. 5. **Using cosine to find hypotenuse:** $$\cos(22^\circ) = \frac{1.7}{c}$$ Rearranged: $$c = \frac{1.7}{\cos(22^\circ)}$$ Calculate: $$\cos(22^\circ) \approx 0.9272$$ $$c \approx \frac{1.7}{0.9272} \approx 1.833$$ cm 6. **Check with Pythagorean theorem:** $$a = c \sin(22^\circ) = 1.833 \times 0.3746 \approx 0.686$$ cm, which does not match the given 3.2 cm vertical leg. 7. **Conclusion:** The given angle and side lengths suggest the triangle is not right-angled at the vertex between the legs 3.2 cm and 1.7 cm. The best approach is to use the Pythagorean theorem for the right triangle with legs 3.2 cm and 1.7 cm. **Final answer:** $$\boxed{3.63 \text{ cm}}$$