lesson
Updated 1 month ago Β· 7 views
You have likely used a2+b2=c2 dozens of times to find missing triangle sides, but have you ever wondered why it is always true? Drawing just one single line inside any right triangle unlocks a stunning, airtight proof.
Around 530 BCE, the Greek philosopher Pythagoras and his followers popularized this relationship, though ancient Babylonian and Indian builders had observed parts of it centuries earlier. Euclidean geometry later unified the concept into an elegant proof using similar triangles.
The Altitude to the Hypotenuse
An altitude is a perpendicular line segment drawn from a vertex to the opposite side. In a right triangle, the hypotenuse is the longest side directly across from the 90β angle.
When you draw an altitude from the right angle straight down to the hypotenuse, it cuts the original large triangle into two smaller right triangles.
πA clean interactive diagram showing a large right triangle ABC with right angle at C and hypotenuse AB at the bottom. An altitude CD descends from C perpendicular to AB, splitting AB into segments AD (length x) and DB (length y). Angle A is colored orange (alpha), angle B is colored blue (beta), and angle C is marked with a 90-degree square. An animation or toggle reveals the three separated triangles: Original ABC, Left triangle ACD, and Right triangle CBD, all labeled with matching color-coded angle arcs (alpha orange, beta blue, 90-degree square) to show they all have the exact same 3 angles.
Because each of the three triangles contains a 90β angle and shares one acute angle with the original triangle, all three are similar triangles by the Angle-Angle (AA) Similarity criterion.
How do these three nested shapes connect side lengths a,b, and c to each other?
Setting Up the Proportions
Let the hypotenuse have total length c, divided by the altitude into two adjacent segments of length x (under leg a) and y (under leg b), such that c=x+y.
Because similar triangles have proportional side lengths, the ratio of a leg to the hypotenuse is the same in the small triangles as it is in the full triangle.
πTwo side-by-side comparison cards. Card 1 (Left): Small triangle ACD compared to Large triangle ABC. Shows ratio: (leg adjacent to alpha) / (hypotenuse) -> x / a = a / c. Arrow points down to cross-multiplication: a^2 = c * x. Card 2 (Right): Medium triangle CBD compared to Large triangle ABC. Shows ratio: (leg adjacent to beta) / (hypotenuse) -> y / b = b / c. Arrow points down to cross-multiplication: b^2 = c * y. Key sides x, y, a, b, c are color-coded consistently.
Cross-multiplying these ratios gives two fundamental geometric equations: a2=cx and b2=cy.