Sheet № 139 · Foundation + Higher · AQA · Edexcel · OCR
Forming Equations from Shapes –
One of the most common exam question styles combines algebra with geometry. You are given a shape with sides or angles written as algebraic expressions and asked to form an equation, then solve it. These questions test whether you can translate geometric rules into algebra. They appear on both Foundation and Higher papers and are worth se
§Key definitions
Question:
A triangle has angles (2x + 10)°, (3x)°, and (x + 50)°. Find the value of x.
Answer:
Two sides of 11 cm and a base of 5 cm
Q1 (Foundation):
A triangle has angles (x + 30)°, (2x + 10)°, and (x + 20)°. Find x and state each angle.
Q2 (Foundation):
A square has side length (2x − 1) cm and perimeter 28 cm. Find x.
Q3 (Higher):
A rectangle has length (5x − 2) cm and width (2x + 1) cm. Its area is 85 cm². Find x.
§Formulas to memorise
Angles in a triangle add up to 180°
Angles in a quadrilateral add up to 360°
Perimeter = sum of all side lengths
For a rectangle, opposite sides are equal
Read the question carefully — and identify which geometric fact applies (angles in a triangle, perimeter, area, etc.).
Write an equation — using the algebraic expressions given in the diagram and the geometric rule.
Simplify the equation — by collecting like terms.
Solve for the unknown — variable.
Substitute back — to find the required measurement (angle, side length, area, etc.).
Check your answer makes sense — side lengths must be positive, angles must be sensible.
Worked example
A triangle has angles (2x + 10)°, (3x)°, and (x + 50)°. Find the value of x.
Working:
⚠ Common mistakes
- ✗Forgetting the geometric rule. Students write the expressions but do not set them equal to anything. You need a fact like "angles sum to 180°" or "perimeter = given value" to form the equation.
- ✗Mixing up perimeter and area. Read the question carefully — perimeter is the total of all sides; area involves multiplication.
- ✗Accepting negative or zero side lengths. If your value of x produces a negative side length, recheck your equation. Side lengths must be positive.
✦ Exam tips
- →Always state the geometric rule you are using — this earns a method mark even if your algebra goes wrong.
- →Show every step of your working. These questions carry 3–5 marks and part marks are available.
- →After finding x, substitute back to answer the specific question asked (the question might ask for a side length or an angle, not x itself).
- →Diagrams in exams are not drawn to scale, so do not estimate from the picture.