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

Constant xy fixes the triangle's area

Find the inverse proportion from P, then let A(p, q) be any point on the first-quadrant branch. Because AB passes through O, B is A reflected through the origin, and the right triangle ABC has legs we can write with p and q. Its area depends only on pq.

Number systemGeometry

Your answer

How to solve

Strategy Change Focus / Count the Complement — Instead of hunting for A, change focus to the product xy, which is the same for every point on the graph. Point symmetry about O turns the legs of the triangle into 2p and 2q.

1STEP 1

Find the constant

For y=k/x, the product xy is the constant k. Use P.

k = (-3) × (-1) = 3 → y = 3/x

2STEP 2

Name the points

Let A=(p, q) with pq=3. The line through O and A is y=(q/p)x, and it meets the curve again at the reflection of A through O.

B = (-p, -q), C = (p, -q)

3STEP 3

Leg lengths

AC is vertical from q down to -q; BC is horizontal from -p to p.

AC = q - (-q) = 2q, BC = p - (-p) = 2p

4STEP 4

Area

Triangle ABC is right-angled at C.

1/2 × 2p × 2q = 2pq = 2 × 3 = 6

The unknown p and q appear only as the product pq, which we know.

7.G.B.6Change Focus Count The Complement

Triangle $ABC$ has area $2pq = 6$ , wherever A sits on the curve.

Why?

B is A turned half a turn about O, so $B = (-p, -q)$ .

Why?

$(-p) \times (-q)$ is a product of like signs, so it equals $pq = 3$ and $(-p, -q)$ is on the curve too.

🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).

Why?

Going from O to $(-p, -q)$ is going from O to A backward, with the same ratio of rise to run, so $(-p, -q)$ is also on line OA.

🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.

Why?

The $x$ -axis cuts AC into $q$ above it and $q$ below it, so $AC = 2q$ , and the $y$ -axis cuts BC the same way, so $BC = 2p$ .

🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

Why?

The triangle is right-angled at C, so its area is $\frac{1}{2} \times 2p \times 2q = 2pq$ .

🧱A triangle is half its base times its heightTwo copies of a triangle fit together into a parallelogram on the same base, so one triangle covers half of base times height.

Why?

$pq = 3$ because A is on $y = \frac{3}{x}$ .

Why?

The curve passes through P, so $k = (-3) \times (-1)$ , a product of two negatives, which is $3$ .

Why?

Multiplying $y = \frac{3}{x}$ back by $x$ undoes the division, so every point on the curve has $xy = 3$ .

🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

Answer

Try a concrete A: A(1, 3) gives B(-1, -3), C(1, -3), legs 2 and 6, area 1/2 × 2 × 6 = 6. A(3, 1) gives legs 6 and 2, again 6.

Takeaway

When a point on y=k/x is not pinned down, the answer usually only needs xy = k.

Find the constant
Name the points
Leg lengths
Area

Where next?

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Count one quadrant, then use point symmetry

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▶ Practice — 12 problems