Geometry: The Pythagorean Theorem

Principle of the Reduced Triangle

1. Reduce the difficulty of the problem by multiplying or dividing the three lengths by the same number to obtain a similar but simpler triangle in the same family 2. Solve for the missing side of this easier triangle 3. Convert back to the original problem

If c is the length of the longest side of a triangle, and

1. a²+b²>c², then the triangle is acute 2. a²+b²=c², then the triangle is right 3. a²+b²<c², then the triangle is obtuse

6 common families of right triangles to memorize

30°-60°-90°⇔(x, x√3, 2x) 45°-45°-90°⇔(x, x, x√2) (3,4,5) (5,12,13) (7,24,25) (8,15,17)

Pythagorean Triples

Any three whole numbers that satisfy the equation a²+b²=c² form a Pythagorean triple. Ex. (3,4,5) (6,8,10 is in the (3,4,5) family) (5,12,13) (7,24,25) (8,15,17) (9,40,41) (11,60,61) (20,21,29) (12,35,37)

Distance Formula/Theorem

If P=(x₁,y₁) and Q=(x₂,y₂) are any two point, then the distance between them can be found with the formula PQ=√[(x₂-x₁)²+(y₂-y₁)²] or PQ=√[(∆x)²+(∆y)²]

Altitude on Hypotenuse Theorem

If an altitude is drawn to the hypotenuse of a right triangle, then 1. The two triangles formed are similar to the given right triangle and to each other. 2. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse (x/h=h/y, or h²=xy) 3. Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg (i.e., the proportion of that leg on the hypotenuse) (y/a=a/c, or a²=yc)(x/b=b/c, or b²=xc)

Angle of Elevation

If an observer at a point p looks upward toward an object at A, the angle the line of sight makes with the horizontal is called the angle of elevation

Converse of the Pythagorean Theorem

If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.

30°-60°-90° Triangle Theorem

In a triangle whose angles have the measures 30, 60, and 90, the lengths of the sides opposite these angles can be represented by x, x√(3), and 2x, respectively

45°-45°-90° Triangle Theorem

In a triangle whose angles have the measures 45, 45, and 90, the lengths of the sides opposite these angles can be represented by x, x, and x√(2), respectively

Pythagorean Theorem

The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs. a²+b²=c²

Angle of Depression

if an observer at a point p looks downward toward an object at B, the angle the line of sight makes with the horizontal is called the angle of depression

Rectangular Solid

•6 rectangular faces •12 edges •d is one of the 4 diagonals •Formula for diagonal: √(L²+h²+w²) Note: A cube is a rectangular solid in which all edges are congruent

Regular Square Pyramid

•Base is a square, and it is called the base •The tip top is called the vertex •h is the altitude of the pyramid and is perpendicular to the base at its center •S is called a slant height and is perpendicular to a side of the base