Square Roots: Pythagoras
Geometry
The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
Suppose we draw two intersecting lines, AB and AC at right angles to each other. If AB = 3 units and AC = 4 units, the triangle ABC is determined and the length of BC is fixed. So the length of AB (called the hypotenuse of the right-angled triangle ABC) must depend on the lengths of the other two sides.
What is the relation which connects the lengths of the three sides of a right-angled triangle?
The relation, first proved rigorously by Pythagoras, a Greek philosopher and geometrician, was known in practice long before his time. It was, in fact, used by the ancient Egyptians for the construction of right angles.
The theorem states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
The hypotenuse BC is therefore 5 units long.
Ex.1. In the triangle, ABC, AC = 4.2 cm and BC = 3.4 cm and angle C is a right angle. Find the length of AB.
AB² = AC² + BC² (Pythagoras)
= 4.2² + 3.4²
= 17.64 + 11.56
= 29.2 cm²
AB = 5.4 cm
Ex. 2. In the triangle ABC, AD is the perpendicular from A to BC. Given that AB = 13 cm, AD = 12 cm and AC = 15 cm, find the length of BC.
AD² + BD² = AB² (Pythagoras)
12² + BD² = AB²
BD² = 13² - 12²
= 25
BD = 5 cm
Also AD² + DC² = 15²
DC² = 15² - 12²
= 81
DC = 9 cm
The length of BC is 5 cm + 9 cm or 14 cm.
Q.3. ABC which is right-angled at C.Given AC= 16 cm, BC = 30 cm find AB.
We are given, AC = 16 cm, BC = 30 cmAB² = AC² + BC² = (16)² + (30)² = 1156 = √1156AB = 34
