In today's geometry lesson, we will show a fairly easy way to prove the perpendicular bisector theorem. A line that splits another line segment (or an angle) into two equal parts is called a … [Read more...] about Perpendicular Bisector Theorem

# Similar Triangles

Two triangles are said to be 'similar' if their corresponding angles are all congruent. Which means they all have the same measure.

This is different from congruent triangles because congruent triangles have the same length and the same angles. But two similar triangles can have the same angles, but with a different size of corresponding side lengths.

Think of it as "expanding" or "shrinking" a triangle by the same amount in all directions, keeping the proportions the same, but the sizes different:

We denote similar triangles using this symbol: ∼, for example:

ΔABC∼ ΔDEF

A key property of similar triangles is that their sides are all proportional with the same ratio, called the "scale factor". So if ΔABC∼ ΔDEF, and side AB is twice as large as side DE, this same scale factor of 2:1 holds for all the other sides as well: BC = 2*EF, AC = 2*DF.

We can write this as AB/DE=BC/EF=AC/DF=scale factor. From this, we can derive other relationships, like the product of multiplying two sides:

AB/DE = BC/EF, so if we cross-multiply we get

AB*EF = BC*DE

And, since we know from basic algebra that if a/b=c/d, then a/c=b/d, we also have AB/BC=DE/EF - so the relationship between two sides of of the triangles is the same as the relationship between those two corresponding triangles in the other similar triangle.

The scale factor also holds true for the triangles' height and perimeter. The areas, which are (base*height/2) is the scale factor squared, since we are multiplying two parts (the base and the height) and each one of them is scaled by the same factor.

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*Now that we've explained the basic concept of similar triangles in geometry, let's scroll down to work on specific geometry problems relating to this topic.*

## Are All Equilateral Triangles Similar?

A regular polygon is one in which all angles are equal and all sides are equal. So by definition, all regular polygons with the same number of sides are similar to each other. An equilateral … [Read more...] about Are All Equilateral Triangles Similar?

## Ratio of Area of Triangles in a Trapezoid

In today's lesson, we will use similar triangles and triangles with the same height to find the ratio between areas of triangles formed by the intersection of a trapezoid's diagonals. The diagonals … [Read more...] about Ratio of Area of Triangles in a Trapezoid

## A Difficult Geometry Problem With Similar Triangles

One of my regular readers sent me the following problem, asking for my help in solving it. This is a challenging problem that was fun to solve, because it uses several concepts we have discussed … [Read more...] about A Difficult Geometry Problem With Similar Triangles

## Calculating the Height of Tall Objects

One application of the properties of similar triangles is to find the height of very tall objects such as buildings using the length of their shadow on the ground and comparing it to the length of the … [Read more...] about Calculating the Height of Tall Objects

## How to Prove that Triangles are Similar

In many of the problems involving similar triangles, you will be asked to prove that the triangles are similar. The easiest way to do this is to show that all the angles are congruent or have an … [Read more...] about How to Prove that Triangles are Similar