The Triangle and its Properties - Right-angled Triangles and Pythagoras Property

Find the length of the hypotenuse of a right-angled triangle whose legs are $6\text{ cm}$ and $8\text{ cm}$ long.

1. Let the legs be $a = 6\text{ cm}$ and $b = 8\text{ cm}$. Let the hypotenuse be $c$.
2. According to Pythagoras Property: $a^2 + b^2 = c^2$
3. Substitute the values: $6^2 + 8^2 = c^2$
4. Calculate the squares: $36 + 64 = c^2$
5. Add the values: $100 = c^2$
6. Find the square root: $c = \sqrt{100} = 10\text{ cm}$.

We use the standard Pythagoras formula where the sum of the squares of the two shorter sides gives the square of the longest side (hypotenuse).

A $13\text{ m}$ long ladder is placed against a wall such that its foot is $5\text{ m}$ away from the wall. At what height does the ladder reach the wall?

1. Here, the ladder acts as the hypotenuse ($c = 13\text{ m}$) and the distance from the wall is the base ($a = 5\text{ m}$).
2. We need to find the height ($b$).
3. Using $a^2 + b^2 = c^2$, we get: $5^2 + b^2 = 13^2$
4. $25 + b^2 = 169$
5. Subtract $25$ from both sides: $b^2 = 169 - 25$
6. $b^2 = 144$
7. $b = \sqrt{144} = 12\text{ m}$.

This is a real-world application of the Pythagoras property where the wall, the ground, and the ladder form a right-angled triangle. We solve for the missing leg (the height on the wall).

1. Identify the longest side: $c = 41\text{ cm}$.
2. Calculate $c^2$: $41^2 = 1681$.
3. Identify the other two sides: $a = 9\text{ cm}, b = 40\text{ cm}$.
4. Calculate $a^2 + b^2$: $9^2 + 40^2 = 81 + 1600 = 1681$.
5. Since $a^2 + b^2 = c^2$ ($1681 = 1681$), the triangle is right-angled.

According to the converse of the Pythagoras property, if the sum of squares of two sides equals the square of the third side, the triangle is right-angled.

1. Let the broken part of the tree be the hypotenuse $c$.
2. The vertical part is $a = 5\text{ m}$, and the distance to the base is $b = 12\text{ m}$.
3. By Pythagoras property: $c^2 = 5^2 + 12^2 = 25 + 144 = 169$.
4. $c = \sqrt{169} = 13\text{ m}$.
5. Original height = Broken part + Vertical part = $13\text{ m} + 5\text{ m} = 18\text{ m}$.

The broken tree forms a right-angled triangle. We find the length of the fallen part using Pythagoras theorem and add it to the standing height.