Key Points

In a right-angled triangle, the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides. If the sides are $a$ and $b$, and the hypotenuse is $c$, then $a^2 + b^2 = c^2$.

Use the theorem $a^2 + b^2 = c^2$ to find a missing side. To find a shorter side $b$, use the formula $b = \sqrt{c^2 - a^2}$. The hypotenuse $c$ is always the longest side of a right triangle.

In an isosceles right triangle with two equal sides of length $a$, the hypotenuse $c$ is given by $c^2 = a^2 + a^2 = 2a^2$. This simplifies to the formula $c = a\sqrt{2}$.

A set of three positive integers $(a, b, c)$ that satisfies the equation $a^2 + b^2 = c^2$ is called a Baudhāyana triple. Common examples are $(3, 4, 5)$, $(5, 12, 13)$, and $(8, 15, 17)$.

A triple is primitive if its numbers have no common factor greater than 1, like $(3, 4, 5)$. If $(a, b, c)$ is a triple, then $(ka, kb, kc)$ for any integer $k > 1$ is a scaled triple, like $(6, 8, 10)$.

To construct a square with double the area of a given square, use the diagonal of the original square as the side of the new square. This is because the square on the diagonal has an area of $2s^2$ if the original side is $s$.

To construct a square with half the area of a given square, join the midpoints of the sides of the original square. The inner square formed has an area that is exactly half of the larger one.

To create a single square whose area is the sum of two different squares with sides $a$ and $b$, construct a right triangle with perpendicular sides $a$ and $b$. The square on its hypotenuse will have the required area of $a^2 + b^2$.

The number $\sqrt{2}$ is irrational, meaning it cannot be written as a fraction $\frac{m}{n}$ where $m$ and $n$ are integers. Its decimal value is non-terminating and non-repeating, with $\sqrt{2} \approx 1.41421...$.

The length of the diagonal $d$ of a square with side length $s$ is found using the theorem: $d^2 = s^2 + s^2 = 2s^2$. The formula is $d = s\sqrt{2}$.

The length of the diagonal $d$ of a rectangle with length $l$ and breadth $b$ is the hypotenuse of the right triangle formed by its sides. The formula is $d = \sqrt{l^2 + b^2}$.

The diagonals of a rhombus bisect each other at right angles. If the diagonals have lengths $d_1$ and $d_2$, the side length $s$ can be found using the formula $s^2 = (\frac{d_1}{2})^2 + (\frac{d_2}{2})^2$.

One method to generate Baudhāyana triples uses the identity $(n-1)^2 + (2n-1) = n^2$. If we choose $n$ such that $2n-1$ is a perfect square, a triple is formed. For example, if $2n-1 = 49 = 7^2$, then $n=25$, giving the triple $(24, 7, 25)$.

While $a^2 + b^2 = c^2$ has infinite integer solutions, Fermat's Last Theorem states that the equation $a^n + b^n = c^n$ has no positive integer solutions for $a, b, c$ if the exponent $n$ is any integer greater than 2.