Depth · one concept from first principles · step 2 of 4

Equal xy at two points fixes the ratio

Put B(a, 0) and C(b, 0) on the x-axis. Then A is directly above B on the curve, D is level with A, and E is the midpoint of BD. Requiring E to be on the curve links b to a; then the x-coordinate of E links a to m.

GeometryExpressions

Your answer

How to solve

Strategy Convert to Algebra — Label every vertex with two letters, a and b. Two points on y=3/x both satisfy xy = 3, which gives one equation between a and b.

1STEP 1

Label the rectangle

A is above B on the curve, so its height is 3/a; D shares that height and sits above C.

A(a, 3/a), B(a, 0), C(b, 0), D(b, 3/a)

2STEP 2

Midpoint E

Average the coordinates of B and D.

E((a+b)/2, 3/(2a))

3STEP 3

E is on the curve

On y=3/x the product xy is 3.

(a+b)/2 × 3/(2a) = 3 → a + b = 4a → b = 3a

4STEP 4

Bring in m

The x-coordinate of E is (a+3a)/2 = 2a, which equals m.

2a = m → a = m/2, so B(m/2, 0)

Answer

(m/2, 0)

Try m=2: B(1, 0), A(1, 3), C(3, 0), D(3, 3), so E= (2, 1.5) and 2 × 1.5 = 3. Both A and E are on the curve, and E's x-coordinate is 2 = m.

Takeaway

Two points on y=k/x: half the height means twice the distance out.

Label the rectangle
Midpoint E
E is on the curve
Bring in m

Where next?

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