Lens 04 of 9

The known fact

Look for a fact you already hold: a Pythagorean triple, a special angle, a pair of conjugate roots, a perfect square.

Some problems are built on a fact and then disguised. A hypotenuse of 25 is the 7-24-25 triple. A root of 2 − √3 with real coefficients brings 2 + √3 with it. tan A = √3 is 60°. Recognise the fact, verify the leftover condition, and the algebra never starts.

The clues that call for it

Quadratic Equations

A hypotenuse or a sum of squares that matches a triple

25 → 7-24-25; 13 → 5-12-13.

Polynomials

An irrational root with real coefficients

2 − √3 is a root, so 2 + √3 is too; read p and q from the sum and product.

Introduction to Trigonometry

A ratio that belongs to a special angle

tan A = √3 → A = 60°.

Circles

A radius and a distance to the centre

Radius 5, OP = 13 → the tangent is 12.

Worked example

A right triangle has hypotenuse 25 and its altitude is 17 less than its base. Find the perimeter.

1
Recognise the fact
25 as a hypotenuse means the 7-24-25 triple.
7² + 24² = 25²
2
Verify the leftover condition
Base minus altitude should be 17.
24 − 7 = 17 ✓
3
Answer
Perimeter = 7 + 24 + 25 = 56. Pythagoras was never applied; it was recalled.

Try the lens

The known fact

The clues that call for it

A right triangle has hypotenuse 13 and its legs differ by 7. What is its perimeter?

Problems that use this lens