Lesson 6-5 Parts of Similar Triangles

Ohio Content Standards:

Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions.

Ohio Content Standards: Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates.

Ohio Content Standards: Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures.

Ohio Content Standards: Apply proportional reasoning to solve problems involving indirect measurements or rates.

Ohio Content Standards: Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence.

Ohio Content Standards: Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects.

Ohio Content Standards: Use proportions in several forms to solve problems involving similar figures (part-to-part, part-to-whole, corresponding sides between figures).

Ohio Content Standards: Use right triangle trigonometric relationships to determine lengths and angle measures.

Ohio Content Standards: Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle measures in similar figures.

Theorem 6.7 Proportional Perimeters Theorem

If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides.

ABC ~ XYZ XZ = 40, YZ = 41, XY = 9, and AC = 9, find the perimeter of ABC. XZ Y 9 C B A

Theorem 6.8

Similar triangles have corresponding altitudes proportional to the corresponding sides.

Theorem 6.8 Q PR A U V W T

Theorem 6.9

Similar triangles have corresponding angle bisectors proportional to the corresponding sides.

Theorem 6.9 X V U T B R Q P

Theorem 6.10

Similar triangles have corresponding medians proportional to the corresponding sides.

Theorem 6.10 Y V U T M R Q P

Theorem 6.11 Angle Bisector Theorem

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

Theorem 6.11 Angle Bisector Theorem D C BA

ABC ~ MNO and BC = 1/3 NO. Find the ratio of the length of an altitude of ABC to the length of an altitude of MNO.

K x L I J In the figure, EFG ~ JKL. ED is an altitude of EFG, and JI is an altitude of JKL. Find x if EF = 36, ED = 18, and JK = 56. E D G F

In the figure, ABC ~ GED. A E FG D C B

Assignment: Pgs evens, evens