1. **State the problem:** Determine which of the statements about points $A(0,0)$, $B(a,0)$, and $C(a,b)$ are true. 2. **Recall the definitions:** - The length of a segment between points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the distance formula: $$\overline{PQ} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - Two vectors are perpendicular if their dot product is zero. 3. **Calculate lengths:** - $\overline{AB}$ is the distance between $A(0,0)$ and $B(a,0)$: $$\overline{AB} = \sqrt{(a-0)^2 + (0-0)^2} = \sqrt{a^2} = a$$ - $\overline{BC}$ is the distance between $B(a,0)$ and $C(a,b)$: $$\overline{BC} = \sqrt{(a - a)^2 + (b - 0)^2} = \sqrt{b^2} = b$$ 4. **Check statement I: $\overline{AB} = \overline{BC}$** - This means $a = b$. - Since $a$ and $b$ are arbitrary, this is not necessarily true. 5. **Check statement II: $\overline{AB} \perp \overline{BC}$** - Vector $\overline{AB} = (a-0, 0-0) = (a,0)$ - Vector $\overline{BC} = (a - a, b - 0) = (0,b)$ - Dot product: $$\overline{AB} \cdot \overline{BC} = a \times 0 + 0 \times b = 0$$ - Since dot product is zero, vectors are perpendicular. - Statement II is true. 6. **Check statement III: $AC = 2a^2 + 2b^2$** - Length $\overline{AC}$ is distance between $A(0,0)$ and $C(a,b)$: $$\overline{AC} = \sqrt{(a-0)^2 + (b-0)^2} = \sqrt{a^2 + b^2}$$ - The statement says $AC = 2a^2 + 2b^2$, which is not equal to $\sqrt{a^2 + b^2}$. - Statement III is false. **Final answer:** Only statement II is true.