Sullivan Algebra and Trigonometry: Section R.3

Geometry Review

Objectives of this Section

• Use the Pythagorean Theorem and Its Converse

• Know Geometry Formulas

A right triangle is on that contains a right angle, that is, an angle of 90°. The side

opposite the right angle is the hypotenuse.

The Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of

the squares of the lengths of the legs.

2 2 2a b c+ =

Example: The Pythagorean Theorem

Show that a triangle whose sides are of lengths 6, 8, and 10 is a right triangle.

We square the length of the sides:

2 2 26 36 8 64 10 100= = =

Notice that the sum of the first two squares (36 and 64) equals the third square (100). Hence the triangle is a right triangle, since it satisfies the Pythagorean Theorem.

Converse of the Pythagorean Theorem

In a triangle, if the square of the length of one side equals the sums of the squares of the lengths of the other two sides, then the triangle is a right triangle. The 90 degree angle is opposite the longest side.

For a rectangle of length L and width W:

2 2Area lw Perimeter l w= = +

For a triangle with base b and altitude (height) h:

Geometry Formulas

1

2Area bh=

For a circle of radius r (diameter d = 2r)2 2Area r Circumference r dp p p= = =

Geometry Formulas

For a rectangular box of length L, width W, and height H:

Volume lwh=

For a sphere of radius r:

3 244

3Volume r Surface Area rp p= =

For a right circular cylinder of height h and radius r:

2Volume r hp=