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Hey there! Ready to discover one of the most famous secrets of geometry? Let's explore how the sides of right-angled triangles are perfectly connected!
A right-angled triangle is a special three-sided shape where two of the sides meet at a perfect 90-degree corner, just like the corner of a textbook or a room. The two sides that form this square corner are called the legs, and the long, slanted side opposite the corner is called the hypotenuse.
The Pythagorean theorem is a simple rule: if you build a square on each of the two shorter sides (legs 'a' and 'b'), their combined area is exactly equal to the area of the square built on the longest side (hypotenuse 'c'). We write this relationship as a² + b² = c².
Let's see this formula in action! If a right triangle has a vertical leg of 3 units and a horizontal leg of 4 units, we can find the hypotenuse by adding their squares: 9 + 16 equals 25. Taking the square root of 25 gives us a hypotenuse of exactly 5 units.
A 10-foot ladder is leaning against a vertical wall. If the bottom of the ladder is parked 6 feet away from the base of the wall, how high up the wall does the ladder reach?
- Identify the parts of our right-angled triangle. The ladder itself is the longest side opposite the ground corner, so the hypotenuse c = 10.
- The distance from the wall along the ground is one of the legs, so we can set a = 6.
- We need to find the height of the wall, which is our missing leg 'b'. Let's write down our formula: a² + b² = c².
- Substitute the values we know into the equation: 6² + b² = 10².
- Simplify the squares: 36 + b² = 100.
- Subtract 36 from both sides to isolate our variable: b² = 64.
- Take the square root of both sides to solve for b: b = √64 = 8. The ladder reaches exactly 8 feet up the wall!
