Year 7 Geometry Worksheets & Practice Tests

Worked examples

Year 7 geometry: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Classifying a triangle by sides and angles

   A triangle has sides of length 5 cm, 5 cm and 6 cm. Classify this triangle by its sides and by its angles.

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   Key idea: Classify by sides first (count equal sides), then by angles (check whether any angle is 90° or greater).

   1. By sides: Two sides are equal (both 5 cm), so the triangle is isosceles.
   2. By angles: Use the cosine rule to find the largest angle, which is opposite the longest side (6 cm):
      $\cos C = \frac{5^2 + 5^2 - 6^2}{2 \times 5 \times 5} = \frac{25 + 25 - 36}{50} = \frac{14}{50} = 0.28$
   3. Since $\cos C = 0.28 > 0$, angle $C$ is acute. The other two angles are equal and also acute (they must each be less than 90° because the triangle's angles sum to 180° and $C$ is already acute with the two remaining angles sharing what is left).
   4. All three angles are acute, so the triangle is acute.

   Answer: Isosceles and acute-angled triangle.

2. Example 2 · Classifying a quadrilateral from its properties

   A quadrilateral has two pairs of parallel sides, all four sides equal in length, and four right angles. Name all the quadrilateral types that this shape belongs to, and state its most specific name.

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   Key idea: Work through the hierarchy of quadrilateral properties to find every type that fits, then choose the most specific.

   1. It has two pairs of parallel sides, so it is a parallelogram.
   2. All four sides are equal, so it is also a rhombus.
   3. It has four right angles, so it is also a rectangle.
   4. A shape that is both a rhombus and a rectangle is a square.
   5. The most specific name is therefore square.

   Answer: It is a parallelogram, rhombus, rectangle, and most specifically a square.

3. Example 3 · Identifying a shape from a real-life description

   A farmer marks out a flat paddock in the shape of a quadrilateral. The paddock has exactly one pair of parallel sides. The parallel sides measure 40 m and 60 m. The other two sides are 35 m and 50 m, and no angles are right angles. What type of quadrilateral is the paddock? State one property that makes it that type and not a parallelogram.

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   Key idea: A quadrilateral with exactly one pair of parallel sides is a trapezium. A parallelogram has two pairs of parallel sides.

   1. Count the pairs of parallel sides: the paddock has exactly one pair, so it cannot be a parallelogram (which needs two pairs).
   2. A quadrilateral with exactly one pair of parallel sides is a trapezium.
   3. The property that distinguishes it from a parallelogram is that only one pair of opposite sides is parallel, not both pairs.

   Answer: The paddock is a trapezium. It has exactly one pair of parallel sides, whereas a parallelogram has two pairs.

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