Similarity in Geometry

Matching shape and scale

Write $\triangle ABC\sim\triangle DEF$ with corresponding vertices in order. For positive side lengths,

Here $k$ maps lengths in $ABC$ to lengths in $DEF$. Reversing the direction of comparison changes the scale factor to $1/k$. Write the direction before starting a proportion so that the numerator does not switch roles halfway through.

Triangles with corresponding sides $6,8,10$ and $9,12,15$ have common ratio $3/2$. The larger triangle’s perimeter is $36$ units; the smaller triangle’s perimeter is $24$ units. The perimeter ratio is also $3/2$ because each side is multiplied by that same factor.

Three ways to establish triangle similarity

AA similarity requires two pairs of equal corresponding angles. The third pair follows from the triangle angle sum. SSS similarity requires all three corresponding side ratios to agree. SAS similarity requires two matching side ratios and equality of the included angles.

Equal angles are powerful for triangles but not sufficient for arbitrary polygons. Every rectangle has four right angles, yet a $2\times3$ rectangle and a $2\times5$ rectangle are not similar. Their corresponding side ratios do not agree.

An AA argument from parallel lines

Let $D$ lie on $AB$ and $E$ on $AC$, with $DE\parallel BC$. The triangles $ADE$ and $ABC$ share the angle at $A$. Corresponding angles formed by the parallel lines give a second equal pair. Therefore

If $AD=4$, $DB=6$ and $AC=15$, then $AB=10$. The small-to-large ratio is $4/10$, so

The remaining segment is $EC=15-6=9$. Confusing $DB$ with the whole side $AB$ would produce a wrong ratio. Mark whether each number describes a whole side or only part of one.

Solving an indirect-measurement problem

Suppose a vertical $1.5\text{ m}$ measuring pole casts a $2\text{ m}$ shadow. At the same time, a vertical object casts an $8\text{ m}$ shadow on the same level surface. Model the sunlight rays as parallel. The two triangles have right angles at the ground and the same angle of sunlight, so they are similar by AA.

Let the unknown height be $h$ metres. Match vertical height to vertical height and shadow to shadow:

The height is $6\text{ m}$. As a reasonableness check, the long shadow is four times the short one, so the corresponding height must also be four times as large.

The model’s assumptions are part of the reasoning. Uneven ground, a tilted pole or measurements taken at different sun angles can invalidate the AA comparison. Similarity cannot repair inconsistent measurements.

Length, area and volume do not scale alike

Lengths and perimeters scale by $k$. Areas scale by $k^2$: in a triangle, both the base and its perpendicular height acquire the factor $k$, so

For similar solids, volumes scale by $k^3$. This volume statement requires similarity in all three dimensions, not just a matching front view.

If the linear factor is $3/2$, the area factor is $9/4$. An original area of $24\text{ cm}^2$ becomes $54\text{ cm}^2$. A volume of $16\text{ cm}^3$ becomes $54\text{ cm}^3$ when the same factor applies in all directions.

Working backwards from area

Two similar figures have areas in the ratio $49:81$. Their corresponding length ratio is $7:9$, because lengths use the positive square root of the area ratio. A $14\text{ cm}$ side of the smaller figure corresponds to $18\text{ cm}$ in the larger one.

Do not take the square root of a perimeter ratio. Perimeter is already a length measure.

Similarity as a transformation

A positive dilation changes size about a fixed centre. Translations, rotations and reflections can then place the enlarged figure in a different position. Together these operations describe similarity in the plane.

A negative coordinate dilation factor is also possible. Its magnitude supplies the positive length scale, while the negative sign places each image on the opposite ray through the centre. Dilation develops this signed convention without treating a length as negative.

Congruence is the case in which the positive length scale is $1$. Similar figures with $k\ne1$ are not congruent even though their angles match.

Conditions that are not enough

Matching one angle and one side does not establish similarity. Nor does matching just two side ratios without controlling the included angle. In a triangle, the chosen test must constrain the entire shape.

A photograph can distort proportions through perspective. Parallel edges can appear to converge, and identical lengths at different depths can appear different. Treat a photograph as an illustration unless a justified geometric model supports a scale calculation. A scale drawing with a stated uniform scale is a different mathematical object.

The AMSI similarity module connects triangle tests with the idea of a constant scale. That connection is the central check: every corresponding length must use the same factor.

Related topics

Dilation makes the scale operation explicit. Regular Polygons uses similar and congruent central triangles, while Pythagorean Theorem connects right-triangle shape with side lengths.