Solving problems using Pythagoras' theorem
Click to explore updated revision resources for GCSE Maths: Solving 2D problems using Pythagoras's, with step-by-step slideshows, quizzes, practice exam questions, and more!
Pythagoras's theorem applies to right-angled triangles. The area of the square drawn on the hypotenuse is equal to the sum of the squares drawn on the other two sides. can be used to solve Having only two dimensions, usually length (or height) and width. problems which involve calculating a length in a right-angled triangle.
Example
Calculate the length UQ. Give the answer to one decimal place.
Draw the right-angled triangle URQ and label the sides.
\(c^2 = a^2 + b^2\)
\(c^2 = 4^2 + 10^2\)
\(c^2 = 16 + 100\)
\(c^2 - 116\)
\(c = \sqrt{116}\)
\(c = 10.8~\text{cm}\)
The length UQ is 10.8 cm (to one decimal place).
It may be necessary to use Pythagoras' theorem more than once in a problem.
Question
Calculate the length S. Give the answer to one decimal place.
It is necessary to calculate the If the angle between two lines is a right angle, the lines are said to be perpendicular. height of the triangles first before calculating S.
First, draw the smaller triangle and label the sides.
\(c^2 = a^2 + b^2\)
\(7^2 = a^2 + 3^2\)
\(49 = a^2 + 9\)
Rearrange the formula to make \(a^2\) the subject.
\(49 - 9 = a^2\)
\(40 = a^2\)
\(a = \sqrt{40}~\text{cm}\)
Draw the larger triangle and label the sides.
\(c^2 = a^2 + b^2\)
\(s^2 = (\sqrt{40})^2 + 6^2\)
\(s^2 = 40 + 36\)
\(s^2 = 76\)
\(s = \sqrt{76}\)
\(s = 8.7~\text{cm}\)
More guides on this topic
- NEW: Angles
- NEW: Angles in parallel lines
- NEW: Polygons
- NEW: Line and rotational symmetry
- NEW: Constructing triangles
- NEW: Ruler and compass constructions
- NEW: Loci
- NEW: Bearings
- NEW: Perimeter
- NEW: Area
- NEW: Volume of a prism
- NEW: Surface area of a prism
- NEW: Pyramids, cones and spheres
- NEW: Nets, plans and elevations
- NEW: Circumference and arc length
- NEW: Area of circles and sectors
- NEW: Higher – Circle Theorems – Calculating angles using circles
- NEW: Higher – Using the alternate segment theorem, tangents and chords
- NEW: Reflection
- NEW: Rotation
- NEW: Translation
- NEW: Enlargement
- NEW: Higher − Negative enlargements
- NEW: Higher – Combined transformations and invariant points
- NEW: Congruent and similar shapes
- NEW: Higher – Similarity in 2D and 3D shapes
- NEW: Pythagoras' theorem
- NEW: Solving 2D and 3D problems using Pythagoras' theorem
- NEW: Right-angled trigonometry
- NEW: Exact trigonometric values
- NEW: Higher – Sine rule
- NEW: Higher – Cosine rule
- NEW: Higher – 2D and 3D trigonometry problems
- NEW: Vectors
- NEW: Higher – Geometric problems using vectors
- Angles, lines and polygons - AQA
- Loci and constructions - AQA
- 2-dimensional shapes - AQA
- 3-dimensional shapes - AQA
- Circles, sectors and arcs - AQA
- Circle theorems - Higher - AQA
- Transformations - AQA
- Units of measure - AQA
- Trigonometry - AQA
- Vectors - AQA
