Learn about the Pythagoras Theorem and how to use it to solve problems involving right-angled triangles.
Example 1: The Pythagoras Theorem
Explain the Pythagoras Theorem.
Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs).
Explanation: If a right-angled triangle has legs of length 'a' and 'b', and the hypotenuse of length 'c', then the theorem states: a² + b² = c².
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Legs: The two shorter sides of a right-angled triangle that form the right angle.
Example 2: Finding the Hypotenuse
A right-angled triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse.
Step 1: Identify the given values. The lengths of the legs are a = 6 cm and b = 8 cm. We need to find the hypotenuse, c.
Step 2: Apply the Pythagoras Theorem: a² + b² = c².
Substitute the given values: 6² + 8² = c².
Step 3: Calculate the squares and add them.
36 + 64 = c²
100 = c²
Step 4: Take the square root of both sides to find c.
c = square root of 100
c = 10
Result: The length of the hypotenuse is 10 cm.
Example 3: Finding a Leg
A right-angled triangle has a hypotenuse of length 13 cm and one leg of length 5 cm. Find the length of the other leg.
Step 1: Identify the given values. The hypotenuse is c = 13 cm, and one leg is a = 5 cm. We need to find the other leg, b.
Step 2: Apply the Pythagoras Theorem: a² + b² = c².
Substitute the given values: 5² + b² = 13².
Step 3: Calculate the squares.
25 + b² = 169
Step 4: Isolate b² by subtracting 25 from both sides.
b² = 169 - 25
b² = 144
Step 5: Take the square root of both sides to find b.
b = square root of 144
b = 12
Result: The length of the other leg is 12 cm.
Example 4: Converse of Pythagoras Theorem
Explain the Converse of the Pythagoras Theorem.
Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
Explanation: In Triangle ABC, if AC² + AB² = BC², then Angle A is a right angle (90 degrees), and Triangle ABC is a right-angled triangle.
Use: The converse is used to determine if a given triangle is a right-angled triangle when you know the lengths of all three sides.
Practice Mode
Enter the lengths of two sides of a right-angled triangle to find the third side.
Note: Enter the sides in the format 'a=value, b=value' or 'a=value, c=value' or 'b=value, c=value', where 'c' is the hypotenuse.
Related Concepts
Explore these related mathematical concepts to deepen your understanding of the Pythagoras Theorem.
Right-Angled Triangle
A triangle with one angle measuring 90 degrees.
Hypotenuse
The side opposite the right angle in a right-angled triangle; it is the longest side.
Legs (of a Right Triangle)
The two sides that form the right angle in a right-angled triangle.
Pythagorean Triple
A set of three positive integers a, b, and c, such that a² + b² = c².
Converse of a Theorem
A statement formed by interchanging the hypothesis and conclusion of a theorem.