1. **State the problem:** Calculate the hypotenuse lengths of right-angled triangles using the Pythagorean theorem and complete the table with opposite, adjacent, and hypotenuse sides. 2. **Recall the Pythagorean theorem:** For a right triangle with sides $a$, $b$ and hypotenuse $r$, the relation is: $$r^2 = a^2 + b^2$$ 3. **Calculate hypotenuse AE:** Given $BE = 4$ (opposite) and $AB = 3$ (adjacent), $$r^2 = 4^2 + 3^2 = 16 + 9 = 25$$ $$r = \sqrt{25} = 5$$ So, $AE = 5$. 4. **Calculate hypotenuse AF:** Given $FC = 8$ (opposite) and $AC = 10 - 5 = 5$ (adjacent), $$r^2 = 8^2 + 5^2 = 64 + 25 = 89$$ $$r = \sqrt{89} \approx 9.43$$ So, $AF \approx 9.43$. 5. **Calculate hypotenuse AH:** Points $A(0,16)$ and $H(16,0)$, so Opposite side $= 16 - 0 = 16$ Adjacent side $= 16 - 0 = 16$ $$r^2 = 16^2 + 16^2 = 256 + 256 = 512$$ $$r = \sqrt{512} = \sqrt{256 \times 2} = 16\sqrt{2} \approx 22.63$$ So, $AH \approx 22.63$. 6. **Complete the table:** | Triangle | Opposite side (y) | Hypotenuse (r) | Adjacent side (x) | |----------|-------------------|----------------|-------------------| | $\triangle ABE$ | $BE = 4$ | $AE = 5$ | $AB = 3$ | | $\triangle AFC$ | $FC = 8$ | $AF \approx 9.43$ | $AC = 5$ | | $\triangle ADH$ | $AH = 16$ | $AH \approx 22.63$ | $AD = 16$ | **Note:** For $\triangle ADH$, adjacent side $AD$ is the horizontal distance from $A(0,16)$ to $D(12,0)$ projected on x-axis, which is $12$; but since $AH$ is hypotenuse, $AD$ is adjacent side along x-axis, so $AD = 12$. Recalculate $AH$ hypotenuse with $AD=12$ and opposite side $DH=16$? Actually, $AH$ is hypotenuse between $A(0,16)$ and $H(16,0)$, so adjacent side $AD$ is $12$ (from $A$ to $D$), opposite side $DH$ is $16$ (from $D$ to $H$). So for $\triangle ADH$: Opposite side $= DH = 16$ Adjacent side $= AD = 12$ $$r^2 = 16^2 + 12^2 = 256 + 144 = 400$$ $$r = \sqrt{400} = 20$$ So, $AH = 20$. Update table: | Triangle | Opposite side (y) | Hypotenuse (r) | Adjacent side (x) | |----------|-------------------|----------------|-------------------| | $\triangle ABE$ | $4$ | $5$ | $3$ | | $\triangle AFC$ | $8$ | $9.43$ | $5$ | | $\triangle ADH$ | $16$ | $20$ | $12$ | **Final answers:** - $AE = 5$ - $AF \approx 9.43$ - $AH = 20$