1. **Problem statement:** Find the length $y$ in each figure based on the given dimensions and geometric properties. --- **a. Kite with horizontal sides 10 cm, 16 cm, 10 cm and vertical sides 5 cm:** 2. The kite is symmetric with two right triangles sharing the vertical side. 3. Half of the horizontal 16 cm side is 8 cm. 4. Use the Pythagorean theorem for one right triangle with legs 5 cm and 8 cm: $$y = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43\text{ cm}$$ --- **b. Triangle with sides 25 cm, 28 cm and height 14 cm:** 5. Use the Pythagorean theorem to find the base segment adjacent to height 14 cm. 6. Let $x$ be the base segment, then: $$x = \sqrt{28^2 - 14^2} = \sqrt{784 - 196} = \sqrt{588} \approx 24.25\text{ cm}$$ 7. The other base segment is $25 - x = 25 - 24.25 = 0.75$ cm. 8. The length $y$ is the height, so $y = 14$ cm. --- **c. Rhombus/kite with sides 10 cm, vertical lengths 5 cm:** 9. The diagonals intersect at right angles, so half diagonals are 5 cm and 5 cm. 10. Use Pythagorean theorem for side length: $$10^2 = 5^2 + y^2 \Rightarrow 100 = 25 + y^2 \Rightarrow y^2 = 75 \Rightarrow y = \sqrt{75} = 5\sqrt{3} \approx 8.66\text{ cm}$$ --- **d. Triangle with sides 26 cm, 7 cm, 10 cm and unknown $y$ with right angles:** 11. The right angles indicate two right triangles sharing side $y$. 12. Use Pythagorean theorem for the triangle with sides 7 cm and $y$: $$y = \sqrt{26^2 - 10^2} = \sqrt{676 - 100} = \sqrt{576} = 24\text{ cm}$$ --- **Final answers:** - a. $y \approx 9.43$ cm - b. $y = 14$ cm - c. $y \approx 8.66$ cm - d. $y = 24$ cm