Problem Solving with Pythagoras Theorem in Geometry
Explore the application of Pythagoras Theorem in solving problems related to right-angled triangles, diagonals of shapes like rectangles and rhombuses, and the height of triangles. Learn how to use Pythagoras Theorem effectively by drawing diagrams, identifying known lengths, and using the theorem to find unknown lengths in various geometric scenarios.
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15 September 2024 Problem solving using Pythagoras Theorem LO: To use Pythagoras theorem in right-angled triangles. www.mathssupport.org
Solving problems using Pythagoras Theorem Pythagoras Theorem can be used to solve many problems involving the lengths in right-angled triangles. When solving a problem using Pythagoras Theorem always do the following: Draw a clearly labelled diagram of the situation. Mark on known lengths and right angles. Use a symbol to represent the unknown length. State Pythagoras Theorem for the given side lengths before substituting any values. Solve the equation. Make sure that your answer is sensible given the lengths of the other two sides. Write your answer in sentence form www.mathssupport.org
Finding the lengths of diagonals Pythagoras Theorem has many applications. For example, we can use it to find the length of the diagonal of a rectangle given the lengths of the sides. Use Pythagoras Theorem to calculate the length of the diagonal, d. d 10.2 cm 13.6 cm d2 =10.22 + 13.62 d2 =104.04 + 184.96 d2 =289 d = 289 d =17 cm www.mathssupport.org
Finding the lengths of diagonals In a square and a rhombus, the diagonals bisect each other at right angles. Use Pythagoras Theorem to calculate the length d of the diagonal in a square of side length 7 cm. d 7 cm Using Pythagoras Theorem d2 = 72 + 72 d2 =49 + 49 d2 =98 d = 98 d =9.90 cm (to 2 d.p.) www.mathssupport.org
Finding the lengths of diagonals A rhombus has diagonals of length 6 cm and 8cm. Find the length of its sides x The diagonals of the rhombus bisect at right angles. Let each side of the rhombus have length x cm. Using Pythagoras Theorem x2 = 42 + 32 x2 =16 + 9 x2 =25 x = 25 x =5 cm www.mathssupport.org
Finding the height of an isosceles triangle In an isosceles triangle and an equilateral triangle, the altitude bisects the base at right angles. isosceles triangle equilateral triangle Use Pythagoras Theorem to calculate the height hof this isosceles triangle. Using Pythagoras Theorem in half of the isosceles triangle, we have 5.8 cm h 4 cm 4 cm 8 cm h2 + 42 = 5.82 h2 =5.82 42 h2 =33.64 16 h2 =17.64 h = 17.64=4.2 cm www.mathssupport.org
Finding the height of an equilateral triangle Use Pythagoras Theorem to calculate the height hof an equilateral triangle with side length 4 cm. 4 cm h 2 cm Using Pythagoras Theorem in half of the equilateral triangle, we have h2 + 22 = 42 h2 =42 22 h2 =16 4 h2 =12 h = 12 h =2 3 exact answer h 3.46 cm (to 3sf) www.mathssupport.org
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