Download MA.912.G.4.5 Apply theorems involving segments divided

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.
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.
There is a lot of situations where
similar triangles naturally arise
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
Proofs using similar triangles
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:
• Refer to the following picture:
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
Secant segments theorem
Theorem: AB*AC=AD*AE
First we form two triangles: ACD and
AEB to help in the proof.
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
Creating illustrations in GeoGebra
as aid in proofs
• Discuss how creating the illustrations can help
to understand the problem and help in the
proof