geometry: similar triangles
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Geometry: Similar Triangles. MA.912.G.4.5 Apply theorems involving segments divided proportionally. Block 28. Similar triangles. Definition: Two triangles are similar if and only if their corresponding angles are congruent and corresponding sides are proportional. - PowerPoint PPT PresentationTRANSCRIPT
Similar triangles
Definition: Two triangles are similar if and only if their
corresponding angles are congruent and corresponding sides are proportional.
Tests for similarity of triangles
(AA Similarity) If two angles of one triangles are congruent to two angles of another triangle, then the triangles are similar.
Tests for similarity of triangles cont.
• Side-Side-Side (SSS) Similarity: If the corresponding side lengths of two triangles are proportional, then triangles are similar
• SAS (SAS) Similarity: It the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent then the triangles are similar
Segments of chords theorem
Theorem: Given AB and CD chords of a circle intersect at point F then AF*FB=CF*FD.
Two-column proof:
Statements:• AB and CD intersect at F• Angle A is congruent to
angle D
• Triangles AFC and DFB are similar
• AF/FD=CF/FB
• AF*FB=CF*FD
Reasoning:• Given• Inscribed angles, and that
intercept the same arc are congruent
• AA similarity principle
• Definition of similar triangles
• Cross products
Secant segments theorem
Theorem: If two secants intersect in an exterior of a circle, then the product of the measures of one secant and its external secant segment is equal to the product of the measures of the other secant and its external secant segment so: AC*AB=AE*DE
Paragraph proof:
AC and AE are secant segments. By the Reflective Property angles BAD and DAB are congruent.
Inscribed angles that intercept the same arcs are congruent. So ACD is congruent to AEB. By the definition of similar triangles, AB/AD=AE/AC.
Since the cross products of a proportion are similar , AB*AC=AD*AE.
Proportional Perimeter Theorem
• If two triangles are similar, then the perimeters are proportional to the measures of the corresponding sides