Geometry and Measurement - The Pythagorean Theorem and its Converse

Parts of a Right-Angled Triangle: A right-angled triangle consists of two shorter sides called legs ($a$ and $b$) and the longest side called the hypotenuse ($c$). Visually, the hypotenuse is always the side directly opposite the $90^{\circ}$ angle, which is typically marked with a small square in the corner.

The Pythagorean Theorem: This theorem states that in any right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. Visually, imagine a triangle with squares attached to each side; the total number of grid units in the two smaller squares equals the units in the largest square.

Relationship of Side Lengths: The relationship is expressed as $a^2 + b^2 = c^2$. This allows us to calculate the length of any side of a right triangle as long as we know the lengths of the other two sides.

Calculating the Hypotenuse: To find the hypotenuse ($c$), you sum the squares of the legs and take the square root of the result ($c = \sqrt{a^2 + b^2}$). Visually, this is the straight-line distance between the non-connected ends of the two perpendicular legs.

Calculating a Leg: To find a missing leg ($a$ or $b$), you subtract the square of the known leg from the square of the hypotenuse and take the square root ($a = \sqrt{c^2 - b^2}$). This operation effectively 'reverses' the theorem to find a shorter side.

The Converse of the Pythagorean Theorem: This is used to determine if a triangle is right-angled. If the side lengths $a, b,$ and $c$ satisfy the equation $a^2 + b^2 = c^2$, then the triangle must contain a right angle. If $a^2 + b^2 \neq c^2$, it is not a right triangle.

Pythagorean Triples: These are sets of three positive integers $(a, b, c)$ that perfectly satisfy the theorem, such as $(3, 4, 5)$ or $(5, 12, 13)$. Visually, these represent triangles with whole-number side lengths that form a perfect $90^{\circ}$ angle.

Distance on a Coordinate Plane: The theorem is used to find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$. By drawing a horizontal line and a vertical line to connect the points, you create a right triangle where the distance is the hypotenuse.