The Baudhayana-Pythagoras Theorem - Hypotenuse of an Isosceles Right Triangle

Find the length of the hypotenuse of an isosceles right triangle whose equal sides are each $6\text{ cm}$ long.

Let the equal sides be $a = 6\text{ cm}$. According to the Baudhayana-Pythagoras theorem: $h^2 = a^2 + a^2$ $h^2 = 6^2 + 6^2$ $h^2 = 36 + 36$ $h^2 = 72$ $h = \sqrt{72} = \sqrt{36 \times 2}$ $h = 6\sqrt{2}\text{ cm}$

Since the triangle is an isosceles right triangle, we substitute the side length into the derived formula $h = a\sqrt{2}$.

If the hypotenuse of an isosceles right triangle is $10\text{ cm}$, find the length of its equal sides.

Let the equal sides be $a$. The hypotenuse $h = 10\text{ cm}$. Using the formula $h^2 = 2a^2$: $10^2 = 2a^2$ $100 = 2a^2$ Dividing by $2$: $a^2 = 50$ $a = \sqrt{50} = \sqrt{25 \times 2}$ $a = 5\sqrt{2}\text{ cm}$

We use the relationship for isosceles right triangles where the square of the hypotenuse is twice the square of one side, then solve for the side length.

Verify the Baudhayana theorem for a triangle with sides $3\text{ cm}$, $3\text{ cm}$, and hypotenuse $\sqrt{18}\text{ cm}$.

The two equal sides are $a = 3$ and $b = 3$. The hypotenuse is $c = \sqrt{18}$. Calculate $a^2 + b^2$: $3^2 + 3^2 = 9 + 9 = 18$ Calculate $c^2$: $(\sqrt{18})^2 = 18$ Since $a^2 + b^2 = c^2$, the theorem is verified.

By squaring the legs and the hypotenuse, we show that the sum of the squares of the legs equals the square of the hypotenuse.

Given side $a = 5\text{ cm}$. Using the formula for isosceles right triangles: $h = a\sqrt{2}$ $h = 5\sqrt{2}\text{ cm}$ Alternatively: $h^2 = 5^2 + 5^2$ $h^2 = 25 + 25 = 50$ $h = \sqrt{50} = 5\sqrt{2}\text{ cm}$ Using $\sqrt{2} \approx 1.414$: $h \approx 5 \times 1.414 = 7.07\text{ cm}$

Since both sides are equal, we apply the Pythagoras theorem where both the base and perpendicular are 5 cm.

Let the length of each equal side be $a$. Given $h = \sqrt{32}\text{ cm}$. Using the formula $h^2 = 2a^2$: $(\sqrt{32})^2 = 2a^2$ $32 = 2a^2$ $a^2 = \frac{32}{2} = 16$ $a = \sqrt{16} = 4\text{ cm}$ Thus, the equal sides are $4\text{ cm}$ each.

We use the relationship between the hypotenuse and the sides of an isosceles right triangle to solve for the side length.