Hint
Compare \(\displaystyle AC^{2}+BC^{2}\) with \(\displaystyle AB^{2}\). In the rectangle ordered A,C,B,D, the opposite vertices A and B form a diagonal.
Worked solution
(a)
The midpoint is \(\displaystyle (\frac{- 3+ 5}{2}, \frac{2+ 8}{2})= (1, 5)\). The coordinate differences are 8 and 6, so \(\displaystyle AB=\sqrt{64+36}=10\).
Answer: Midpoint \((1,5)\); \(AB=10\).
(b)
\(\displaystyle AC^{2}=7^{2}+(-1)^{2}=50\) and \(\displaystyle BC^{2}=(-1)^{2}+(-7)^{2}=50\). Their sum is \(\displaystyle AB^{2}=100\), so the converse of Pythagoras gives a right angle between CA and CB, at C.
Answer: \(AC^2=BC^2=50\) and \(AB^2=100\).
(c)
AB is a diagonal, and the two diagonals share midpoint (1,5). Hence \(\displaystyle D=2(1,5)-C=(-2,9)\). The parallelogram has the right angle already established at C, so it is a rectangle.
Answer: \(D=(-2,9)\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Give midpoint (1,5).
- 1 mark: Use the correct differences in the distance formula.
- 1 mark: Obtain \(\displaystyle AB=10\).
Alternative methods: A correct vector-length calculation is accepted.
Common error: Using sums instead of differences for the length.
(b) · 3 marks
- 1 mark: Calculate \(\displaystyle AC^{2}=50\) and \(\displaystyle BC^{2}=50\).
- 1 mark: Compare their sum with \(\displaystyle AB^{2}=100\).
- 1 mark: Identify C as the right-angle vertex with a valid justification.
Alternative methods: The dot product \(\displaystyle CA\cdot CB=(-7)(1)+(1)(7)=0\), or a valid perpendicular-gradient argument, also earns full credit.
Common error: Identifying the endpoint of the hypotenuse as the right-angle vertex.
(c) · 2 marks
- 1 mark: Use the shared diagonal midpoint or \(\displaystyle D=A+B-C\).
- 1 mark: Give \(\displaystyle D=(-2,9)\) with the specified vertex order.
Alternative methods: A parallel-side coordinate construction is accepted.
Common error: Treating AB as a side rather than the specified diagonal.
Total: 8 marks.