Learn about similar figures and the criteria for proving triangle similarity.
Example 1: What are Similar Figures?
Explain the concept of similar figures, focusing on triangles.
Concept: Two figures are said to be similar if they have the same shape but not necessarily the same size.
For Polygons (like triangles): Two polygons are similar if:
Their corresponding angles are equal.
Their corresponding sides are in the same ratio (proportional). This ratio is called the scale factor.
Notation: The symbol '$\sim$' is used for similarity. For example, Triangle ABC $\sim$ Triangle PQR means Triangle ABC is similar to Triangle PQR.
Example 2: AAA Similarity Criterion
Explain the Angle-Angle-Angle (AAA) similarity criterion.
Criterion: If the corresponding angles of two triangles are equal, then the triangles are similar.
Explanation: If in Triangle ABC and Triangle PQR, Angle A = Angle P, Angle B = Angle Q, and Angle C = Angle R, then Triangle ABC $\sim$ Triangle PQR by AAA similarity.
Note: If two angles of one triangle are respectively equal to two angles of another triangle (AA similarity), then the third angles must also be equal (since the sum of angles in a triangle is 180 degrees). So, AA similarity is sufficient to prove AAA similarity.
Example 3: SSS Similarity Criterion
Explain the Side-Side-Side (SSS) similarity criterion.
Criterion: If the corresponding sides of two triangles are in the same ratio (proportional), then the triangles are similar.
Explanation: If in Triangle ABC and Triangle PQR, AB/PQ = BC/QR = CA/RP, then Triangle ABC $\sim$ Triangle PQR by SSS similarity.
Example: If Triangle ABC has sides 3, 4, 5 and Triangle PQR has sides 6, 8, 10, then 3/6 = 4/8 = 5/10 = 1/2. The sides are proportional, so the triangles are similar.
Example 4: SAS Similarity Criterion
Explain the Side-Angle-Side (SAS) similarity criterion.
Criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are in the same ratio (proportional), then the two triangles are similar.
Explanation: If in Triangle ABC and Triangle PQR, Angle A = Angle P and AB/PQ = AC/PR, then Triangle ABC $\sim$ Triangle PQR by SAS similarity.
Example 5: Basic Proportionality Theorem (BPT)
Explain the Basic Proportionality Theorem (Thales Theorem).
Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
Explanation: In Triangle ABC, if a line DE is drawn parallel to BC, intersecting AB at D and AC at E, then AD/DB = AE/EC.
Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Practice Mode
Enter problem details to check for similarity or solve using BPT.
Note: This solver handles basic SSS similarity checks and BPT problems (solving for a missing segment).
Related Concepts
Explore these related mathematical concepts to deepen your understanding of Similarity.
Congruence
Figures that have the same shape and the same size.
Ratio
A comparison of two quantities by division.
Proportion
An equation stating that two ratios are equal.
Corresponding Angles
Angles in the same relative position in similar figures.
Corresponding Sides
Sides in the same relative position in similar figures.
Scale Factor
The ratio of corresponding side lengths of two similar figures.
Basic Proportionality Theorem (BPT)
Theorem about a line parallel to one side of a triangle.
Converse of BPT
The inverse statement of the Basic Proportionality Theorem.