High School Algebra Archives - Page 11 of 31

A Detailed Derivation of the Heron’s Formula

June 9, 2013 GB High School Algebra, High School Mathematics

Heron’s Formula can be used to find the area of a triangle given the lengths of the three sides. A triangle with side lengths $a$ , $b$ , and $c$ , its area $A$ can be calculated using the Heron’s formula

$A = \sqrt{s(s-a)(s-b)(s-c)}$

where

$s = \displaystyle \frac{a+b+c}{2}$

is the semiperimeter (half the perimeter) of the triangle.

In this post, I will provide a detailed derivation of this formula. » Read more

2 comments area of triangle, cosine law, heron's formula, heron's formula derivation, heron's formula proof, pythagorean identity

The Geometry of the Least Common Denominator

March 23, 2013 GB Elementary School Number Theory, High School Algebra, High School Mathematics

In mathematics, in putting together things, we must have a commonality; we must add objects that belong to the same set. We add 2 apples and 4 apples to get 6 apples. We do not add apples and oranges and come up with a single- kind-of-fruit-sum.

This is also true with numbers and measurements: we add, subtract, multiply or divide numbers that belong to the same set or measures with the same unit. We do not add binary numbers to decimal numbers and get a result without conversion. We must either convert binary numbers to decimal, or vice versa and then perform addition or any other operations. Also, in finding the area of a rectangle with length 10 inches and width 5 centimeters, the answer must either be in square inches or in square centimeters. » Read more

One comment adding dissimilar fractions, common denominator, getting the lcd, least common denominator, least common multiple

Finding the Sum of the Arithmetic Sequence

February 3, 2013 GB High School Algebra, High School Mathematics

An arithmetic sequence is a sequence of numbers such that the difference between two consecutive terms is constant. The sequence

7, 13, 19, 25, 31, 37, 43, 59

is an example of an arithmetic sequence with first term 7, constant difference 6, and last term 49.

You have learned in that the formula for finding the nth term of the arithmetic sequence $a_n$ with first term $a_1$ , and constant difference $d$ is given by

$a_n = a_1 + (n-1)d$ .

In this post, we derive the formula for finding the sum of all the numbers in an arithmetic sequence. We take the specific example above and use Gauss’ method in finding the sum of the first 100 positive integers. Recall that in adding the first 100 integers, Gauss added the first integer to the last, the second integer to the second to the last, the third integer and the third to the last and so on. » Read more

4 comments arithmetic progression, arithmetic sequence, derivation of sum of arithmetic sequence, sum of arithmetic sequence, sum of arithmetic sequence formula

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