Porisma

Worked solution

Use equal vectors and diagonal lengths to identify a quadrilateral

Referred to an origin OO, the position vectors of four non-collinear points AA, BB, CC and DD are a\mathbf a, b\mathbf b, c\mathbf c and d\mathbf d respectively. Given that a−b=d−c\mathbf a-\mathbf b=\mathbf d-\mathbf c, show that ABCDABCD is a parallelogram.

Given also that ∣a−c∣=∣b−d∣|\mathbf a-\mathbf c|=|\mathbf b-\mathbf d|, identify the shape of the parallelogram ABCDABCD, justify your answer.

[N2004/I/3]

CJC · Discussion · Question 2

Answer

ABCDABCD is a parallelogram; the equal-diagonal condition makes it a rectangle.

Show the parallelogram property

a−b=BA→\mathbf a-\mathbf b=\overrightarrow{BA} and d−c=CD→\mathbf d-\mathbf c=\overrightarrow{CD}. Their equality means these opposite sides have the same length and direction. Hence one pair of opposite sides is equal and parallel, so the nondegenerate quadrilateral ABCDABCD is a parallelogram.

How did this part go?

Use the extra magnitude condition

∣a−c∣=∣b−d∣|\mathbf a-\mathbf c|=|\mathbf b-\mathbf d| says CA=DBCA=DB: the diagonals of the parallelogram have equal lengths. A parallelogram with equal diagonals is a rectangle. The comparison figure illustrates this final special case; the printed generic parallelogram sketch supplies the same vertex and arrow relationships.

How did this part go?

Compare your sketch

Expand diagram

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Rectangle ABCD, with A bottom left, B bottom right, C top right, D top left. Arrows B to A and C to D show equal opposite-side vectors. Both dashed diagonals are drawn. The rectangle illustrates the final equal-diagonal case.
Rectangle ABCD, with A bottom left, B bottom right, C top right, D top left. Arrows B to A and C to D show equal opposite-side vectors. Both dashed diagonals are drawn. The rectangle illustrates the final equal-diagonal case.
Rectangle ABCD, with A bottom left, B bottom right, C top right, D top left. Arrows B to A and C to D show equal opposite-side vectors. Both dashed diagonals are drawn. The rectangle illustrates the final equal-diagonal case.

Common mistakes

The second given condition equates magnitudes, not the two diagonal vectors. Losing the bars changes the mathematical problem.

Solution notes

The printed sketch is a generic slanted parallelogram; the independent diagram illustrates the final rectangle while retaining vertex order, both diagonals and B→A/C→D arrows. No lengths are inferred from the printed drawing.

Read the hint