Use the Pythagorean theorem to find side lengths, areas, and diagonals of plane figures.
📘 Example ① (Diagonal of a Rectangle)
Find the length of the diagonal of a rectangle with length 5cm and width 12cm.
Answer: c² = 5² + 12² = 25 + 144 = 169. c = 13cm
Find the length of the diagonal of a rectangle with length 5cm and width 12cm.
Answer: c² = 5² + 12² = 25 + 144 = 169. c = 13cm
📘 Example ② (Finding Area)
Find the area of an isosceles triangle with base 8cm and equal sides 10cm.
Answer: Let h be the altitude from the vertex to the base: h² = 10² - 4² = 100 - 16 = 84. h = 2√21.
Area = (1/2) × 8 × 2√21 = 8√21 cm²
Find the area of an isosceles triangle with base 8cm and equal sides 10cm.
Answer: Let h be the altitude from the vertex to the base: h² = 10² - 4² = 100 - 16 = 84. h = 2√21.
Area = (1/2) × 8 × 2√21 = 8√21 cm²
💡 Key Point
- If the height is unknown → find it using the Pythagorean theorem
- Draw an auxiliary line (a perpendicular) to form a right triangle
Practice Problems
- Find the diagonal of a rectangle with length 8cm and width 6cm.
- Find the area of an equilateral triangle with side length 6cm (leave the answer in terms of √3).
Answers and Explanations
- Answer: 10cm. 6²+8²=100=10².
- Answer: 9√3 cm². Height=3√3. Area=(1/2)×6×3√3=9√3.