1. **State the problem:** We have a quadrilateral with sides 12 cm (bottom), 5 cm (right vertical), 15 cm (slanted right upper side), and two unknown lengths $x$ (diagonal from bottom-left to right-mid) and $y$ (left side from bottom-left to top). There are right angles at the bottom-right corner and at the right-mid corner between $x$ and the 15 cm side. 2. **Identify the right triangles:** The right angle at the bottom-right corner means the bottom side (12 cm) and the right vertical side (5 cm) are perpendicular. 3. **Use the Pythagorean theorem for the bottom-right triangle:** The diagonal $x$ connects bottom-left to right-mid, forming a right triangle with legs 12 cm and 5 cm. Formula: $$x = \sqrt{12^2 + 5^2}$$ Calculate: $$x = \sqrt{144 + 25} = \sqrt{169} = 13$$ 4. **Use the right angle at right-mid corner:** The 15 cm side is the hypotenuse of a right triangle with legs $x=13$ cm and $y$ (unknown). Formula: $$15^2 = x^2 + y^2$$ Substitute $x=13$: $$225 = 169 + y^2$$ Solve for $y^2$: $$y^2 = 225 - 169 = 56$$ Calculate $y$: $$y = \sqrt{56} = 2\sqrt{14} \approx 7.48$$ 5. **Final answers:** - $x = 13$ cm - $y \approx 7.48$ cm These lengths satisfy the right angle conditions and side lengths given.