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
DefinitionTwo 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.
Angle-Angle (AA) Similarity TheoremIf 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. Solution to Example 1
Side-Side-Side (SSS) Similarity TheoremIf 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 TheoremIf 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. Solution to Example 3
Similar Triangles Problems with SolutionsProblems 1In the triangle ABC shown below, A'C' is parallel to AC. Find the length y of BC' and the length x of A'A. Solution to Problem 1
Problems 2
Solution to Problem 2
Problems 3
Solution to Problem 3
Problems 4
Solution to Problem 4
Problems 5
Solution to Problem 5
More References and Links to Geometry ProblemsIntercept Theorem and Problems with SolutionsGeometry Tutorials, Problems and Interactive Applets Congruent Triangles Examples |