Mathematics 10 — Exercise Review · Q5

Full question

Garden corners are A(2,1),B(4,4),C(1,5). Can a circular fountain pass through all three?

Step-by-step solution

1. Compute the direction changes AB=(4−2,4−1)=(2,3) and AC=(1−2,5−1)=(−1,4).

2. Calculate this operation before combining the remaining terms.

   $$2\times 4=8$$

3. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times \left(-3\right)=3$$

4. Calculate this operation before combining the remaining terms.

   $$8+3=11$$

5. The determinant 2×4−3×(−1)=8+3=11 is nonzero, so the points are non-collinear.

   $$2\times 4--3=11$$

6. Exactly one circle passes through any three non-collinear points.

Answer

Yes; a unique circle exists.