Understanding Line Symmetry

Many people believe that symmetry is beauty. Nature is full of symmetric objects. There are many man-made structures that are also symmetric. In this post, we are going to discuss some of the basic mathematical properties of symmetric objects. We will limit our discussion to line symmetric objects.

Line Symmetry

A figure is line symmetrical when it can be folded along a straight line such that the folded shapes fit exactly on top of each other. The fold line is called the line of symmetry.

When a symmetric figure is folded along its line of symmetry, the parts that are on top of each other are called the corresponding parts. In the polygon below with line of symmetry AB, points C and D are corresponding points, segments GB and HB are corresponding sides, and angle G and angle H are corresponding angles. Since the folded shapes fit exactly on top of each other, the corresponding angles are congruent and their corresponding sides are also congruent.  » Read more

Derivation of the Area of a Rhombus

A rhombus is a parallelogram whose sides are congruent. The diagonals of rhombus are perpendicular to each other. They also bisect each other. In this post, we are going to find the general formula for finding the area of a rhombus using these properties. We are going to learn two methods.

Method 1

Consider the rhombus below.

area of a rhombus

We can divide it into two congruent triangles using diagonal d_1. Since the diagonals of a rhombus are perpendicular to each other, we can use d_1 as base and one half of d_2 as the height of the upper triangle (Why?). If we let A_T be the area of the upper triangle, then, calculating its area, we have  » Read more

Proof Without Words: If x >0, x + 1/x >=2

A picture is worth a thousand words, even in mathematics. Most of the time, complicated algebraic statements can be shown using geometric representations. One example is the sum of

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots

which when added equals 1.

In this post, we show another geometric proof without words. Proof without words is not a proof per se but rather a representation that will help readers understand the proof.  » Read more

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