Properties of Similar Triangles

Two triangles are called similar when they have the same shape, even if their sizes are different.

This happens when:

• All corresponding angles are equal, and
• All corresponding sides are in the same ratio.

△ABC  ∼  △DEF
(same shape, different size)

If two triangles are similar, each angle in one triangle matches an equal angle in the other.

So if:

• \(\angle A = \angle D\)
• \(\angle B = \angle E\)
• \(\angle C = \angle F\)

then the triangles are similar.

If triangles are similar, the lengths of their corresponding sides are always in the same ratio.

In triangles \(\triangle ABC\) and \(\triangle DEF\):

\( \dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD} \)

Side ratio stays constant → similar triangles

Since all corresponding sides are proportional, the perimeters are also in the same ratio.

\( \dfrac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle DEF} = \dfrac{AB}{DE} \)

This helps compare overall sizes of similar triangles.

For similar triangles, the ratio of their areas is equal to the square of the ratio of corresponding sides.

\( \dfrac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\dfrac{AB}{DE}\right)^2 \)

This happens because area depends on the square of length.

In similar triangles, not just sides but all corresponding linear measurements follow the same ratio. This includes heights.

\( \dfrac{h_1}{h_2} = \dfrac{AB}{DE} \)

Heights drop from corresponding vertices.

Medians, angle bisectors, and even perpendicular distances from corresponding vertices are all proportional to the side ratio.

So if the side ratio is \(k\), then:

• Median ratio = \(k\)
• Angle bisector ratio = \(k\)
• Altitude ratio = \(k\)

The properties of similar triangles help you:

• Find unknown lengths easily,
• Solve height and shadow problems,
• Work with maps and scale drawings,
• Understand trigonometry concepts,
• Solve geometry problems efficiently.

Similarity is a powerful tool in mathematics and real-world measurements.