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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:

Similar Triangles

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.

perpendicular bisector

Perpendicular Bisector Theorem

Equilateral triangles

Are All Equilateral Triangles Similar?

trapezoid with diagonals

Ratio of Area of Triangles in a Trapezoid

harder geometry problem with similar triangles

A Difficult Geometry Problem With Similar Triangles

Geometry drawing: objects and their shadows

Calculating the Height of Tall Objects

Similar triangles in geometry

How to Prove that Triangles are Similar

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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