Write one equation for each condition
A phrase such as “equidistant from two points” is already an algebraic instruction. With coordinates, compare squared distances to avoid unnecessary square roots. In a triangle, identify which sides and angles are known before choosing a trigonometric rule.
- Draw and label the configuration, including all possible positions allowed by the question.
- Translate each condition independently. Choose coordinates or a geometric theorem that simplifies the relationships.
- Check that every resulting point or angle satisfies all the original conditions. A plausible sketch alone is not a proof.
Worked example
A locus meeting a circle
A point P = (x, y) is equidistant from A = (−2, 0) and B = (4, 0). It also lies on x² + y² = 10. Find all possible positions of P.
Equal squared distances give (x + 2)² + y² = (x − 4)² + y². Cancel y² and expand:
x² + 4x + 4 = x² − 8x + 16.
Hence 12x = 12, so x = 1. This is the perpendicular bisector of AB, which also follows directly from the midpoint (1, 0).
Substitution into the circle equation gives y² = 9. Thus P = (1, 3) or (1, −3). Both have squared distance 18 from A and B, and both lie on the circle.
Your turn
Two sides and the included angle
A triangle has sides of lengths 5 and 7 enclosing an angle of 60°. Find the third side and the exact area.
Show a hint
The included angle gives the third side by the cosine rule and the area by one half times the two sides times the sine of their included angle.
Show the full solution
By the cosine rule, the third side c satisfies c² = 5² + 7² − 2 × 5 × 7 × cos 60° = 25 + 49 − 35 = 39. As a length, c = √39.
The area is (1/2) × 5 × 7 × sin 60° = (35/2)(√3/2) = 35√3/4 square units.
The third side lies between 2 and 12, as required by the triangle inequality. Keep exact trigonometric values until the end.
Explain why the two questions call for different tools. Which words in each question told you what equation to write?
What to practise next
Practise coordinates and loci first, then triangles. On a mixed question, explain your choice of method before calculating.
- Physics & Maths Tutor: coordinate geometry
Use straight-line and circle questions, checking all intersections.
- Physics & Maths Tutor: trigonometry
Practise exact values and triangle rules with the matching solutions.
- Interview practice problems
Practise explaining how the geometry and algebra fit together.
The examples and explanations on this page are part of Arij Asad’s teaching guides. Linked papers and worksheets belong to their named creators and publishers. PMT’s topic collections include exam-board questions and other credited materials.