Download Lesson 49

NYS COMMON CORE MATHEMATICS CURRICULUM
Name: _____________________________
LESSON 49:
Lesson 49
M2
GEOMETRY
Date: ______________
SPECIAL RELATIONSHIPS WITHIN RIGHT TRIANGLES-DIVIDING INTO TWO
SIMILAR SUB-TRIANGLES
OBJECTIVE: SWBAT discover that the altitude of a right triangle from the vertex
of the right angle to the hypotenuse divides the triangle into two
similar right triangles that are also similar to the original right
triangle.
OPENING EXERCISE
Use the diagram to complete
parts (a)–(c).
a. Are the triangles similar?
Explain.
b. Determine the unknown lengths of the triangles.
c. Explain how you found the lengths in part (b).
Page 1 of 8
Lesson 49
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
GEOMETRY
Example 1
Recall that an altitude of a triangle is a perpendicular line segment from a vertex
̅̅̅̅ is the
to the line determined by the opposite side. In triangle △ 𝐴𝐵𝐶 below, 𝐵𝐷
̅̅̅̅ .
altitude from vertex 𝐵 to the line containing 𝐴𝐶
How many triangles do you see in the figure? ____
Re-draw all three right triangles so that they are
facing the same way.
Identify the three right triangles by name.
*Note that there are many ways to name the three triangles. Name the triangles
to show the corresponding angles.
________
Page 2 of 8
________
________
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 49
M2
GEOMETRY
̅̅̅̅ divides the right triangle into two sub-triangles, △ 𝐵𝐷𝐶
In △ 𝐴𝐵𝐶, the altitude 𝐵𝐷
and △ 𝐴𝐷𝐵.
Is △ 𝐴𝐵𝐶~ △ 𝐵𝐷𝐶? Is △ 𝐴𝐵𝐶~ △ 𝐴𝐷𝐵? Explain.
Since △ 𝐴𝐵𝐶 ~ △ 𝐵𝐷𝐶 and △ 𝐴𝐵𝐶~ △ 𝐴𝐷𝐵, can we conclude that △ 𝐵𝐷𝐶~ △ 𝐴𝐷𝐵?
Explain.
Page 3 of 8
Lesson 49
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
GEOMETRY
Example 2
Identify the altitude drawn in
triangle △ 𝐸𝐹𝐺.
_________
Re-draw all three right triangles so that they are facing the same way.
Identify all three triangles by name so that the corresponding angles match up.
_____________
_____________
_____________
Identify all corresponding sides of all three right triangles.
Shorter Legs: _____________
_____________
_____________
Longer Legs:
_____________
_____________
_____________
Hypotenuses: _____________
_____________
_____________
**Since the triangles are similar, the ratios of their corresponding sides will be
equal.
Page 4 of 8
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 49
M2
GEOMETRY
Example 3
Use similar triangles to find the length of the altitude labeled with a variable in
the triangle below.
Page 5 of 8
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 49
M2
GEOMETRY
Example 4
̅̅̅̅ is drawn to hypotenuse
In the diagram below of right triangle 𝐴𝐵𝐶, altidude 𝐵𝐷
̅̅̅̅ , 𝐴𝐶 = 16, and 𝐶𝐷 = 7. Find the length of 𝐵𝐷
̅̅̅̅ in simplest radical form.
𝐴𝐶
Page 6 of 8
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 49
M2
GEOMETRY
Problem Set
1. Use similar triangles to find the length of the altitude labeled with a variable
in the triangle below.
Page 7 of 8
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 49
M2
GEOMETRY
2. In the diagram below of right triangle 𝐴𝐶𝐵, altitude ̅̅̅̅
𝐶𝐷 intersects ̅̅̅̅
𝐴𝐵 at 𝐷. If
𝐴𝐷 = 3 and 𝐷𝐵 = 4, find the length of ̅̅̅̅
𝐶𝐷 in simplest radical form.
Page 8 of 8