Similar Triangles Examples and Problems with Solutions

Definitions and theorems related to similar triangles are discussed using examples. Also examples and problems with detailed solutions are included.

Review of Similar Triangles

Definition

Two triangles ABC and A'B'C' are similar if the three angles of the first triangle are congruent to the corresponding three angles of the second triangle and the lengths of their corresponding sides are proportional as follows.
AB / A'B' = BC / B'C' = CA / C'A'

definition of similar triangles.

Angle-Angle (AA) Similarity Theorem

If two angles in a triangle are congruent to the two corresponding angles in a second triangle, then the two triangles are similar.
Example 1
Let ABC be a triangle and A'C' a segment parallel to AC. What can you say about triangles ABC and A'BC'? Explain your answer.
Angle-Angle similarity

Solution to Example 1

Side-Side-Side (SSS) Similarity Theorem

If the three sides of a triangle are proportional to the corresponding sides of a second triangle, then the triangles are similar.
Example 2
Let the vertices of triangles ABC and PQR defined by the coordinates: A(-2,0), B(0,4), C(2,0), P(-1,1), Q(0,3), and R(1,1). Show that the two triangles are similar.

Solution to Example 2

Side-Angle-Side (SAS) Similarity Theorem

If an angle of a triangle is congruent to the corresponding angle of a second triangle, and the lengths of the two sides including the angle in one triangle are proportional to the lengths of the corresponding two sides in the second triangle, then the two triangles are similar.
Example 3
Show that triangles ABC and A'BC', in the figure below, are similar.
Side-Angle-Side (SAS) similarity

Solution to Example 3

Similar Triangles Problems with Solutions

Problems 1
In the triangle ABC shown below, A'C' is parallel to AC. Find the length y of BC' and the length x of A'A.
similar triangles problem 1

Solution to Problem 1

Problems 2
A research team wishes to determine the altitude of a mountain as follows (see figure below): They use a light source at L, mounted on a structure of height 2 meters, to shine a beam of light through the top of a pole P' through the top of the mountain M'. The height of the pole is 20 meters. The distance between the altitude of the mountain and the pole is 1000 meters. The distance between the pole and the laser is 10 meters. We assume that the light source mount, the pole and the altitude of the mountain are in the same plane. Find the altitude h of the mountain.

altitude of a mountain problem 2

Solution to Problem 2

Problems 3
The two triangles are similar and the ratio of the lengths of their sides is equal to k: AB / A'B' = BC / B'C' = CA / C'A' = k. Find the ratio BH / B'H' of the lengths of the altitudes of the two triangles.

altitude of similar triangles problem 3

Solution to Problem 3

Problems 4
BA' and AB' are chords of a circle that intersect at C. Find a relationship between the lengths of segments AC, BC, B'C and A'C.

intersecting chords inside a circle problem 4

Solution to Problem 4

Problems 5
ABC is a right triangle. AM is perpendicular from vertex A to the hypotenuse BC of the triangle. How many similar triangles are there?

similar right triangles problem 5

Solution to Problem 5

More References and Links to Geometry Problems

Intercept Theorem and Problems with Solutions
Geometry Tutorials, Problems and Interactive Applets
Congruent Triangles Examples