Solid Geometry and the Pythagorean Theorem

Right Triangles, the Pythagorean Theorem, and Distance

A right triangle contains one 90-degree angle. The two shorter sides that intersect to form this 90-degree angle are called the legs. Directly opposite that 90-degree angle sits the hypotenuse, which is always the longest side of a right triangle. The relationship between these sides is defined by the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the leg lengths and c is the hypotenuse length.

You will often encounter real-world situations modeled by right triangles. For example, a ladder leaning against a vertical wall forms a right triangle with the wall and the flat ground. Similarly, a diagonal pathway across a rectangular field forms a right triangle with two adjacent outer edges. To solve for an unknown hypotenuse, add the squares of the two legs and calculate the square root of the sum. To solve for an unknown leg, subtract the square of the known leg from the square of the hypotenuse and take the square root.

On the Cartesian plane, the distance between two coordinate points (x1, y1) and (x2, y2) is given by d = sqrt((x2 - x1)^2 + (y2 - y1)^2). This coordinate distance formula is an algebraic application of the Pythagorean theorem. The horizontal distance between two coordinate points is the absolute difference between their x-values, and the vertical distance is the absolute difference between their y-values. Squaring the coordinate differences makes each term nonnegative, so the distance cannot be negative; it is zero when the points are identical. Geometric distance on a coordinate plane is always a non-negative value.

You can easily calculate these values using your handheld or on-screen calculator. On the TI-30XS MultiView calculator, the x² key squares a number, and pressing 2nd then x² accesses the square-root function.