Download Mod 2 - Aim #22 - Manhasset Public Schools

CC Geometry H
Aim #22: What ratios are applied to right triangles in the study of trigonometry?
Do Now: Trigonometry is applied only to RIGHT triangles.
The label "opposite" will refer to a leg opposite a designated acute angle.
The label "adjacent" will refer to a leg adjacent to a designated acute angle.
0
The label "hypotenuse" will always be the hypotenuse/side across from 90 .
a) Name the side opposite ≮A _____
A
b) Name the side opposite ≮B _____
c) Name the hypotenuse. _____
d) Name the side adjacent to ≮A _____
e) Name the side adjacent to ≮B _____
B
C
Label the three sides as adjacent, opposite, or hypotenuse with respect to the
marked acute angle.
Q
F
D
R
E
K
J
P
Exploratory Challenge:
-Each half of the class gets a set of triangles along with a table.
-Complete your table for missing angle measures and side lengths using a ruler
and protractor when needed. Some cells have been completed. In the "angle
measures" column, you must determine and measure which acute angle is being
used.
-Calculate the ratios adj/hyp and opp/hyp to two decimal places.
Let's share the results.
1) What can you conclude about each pair of triangles between the two sets?
2) What do you notice about the ratios of the corresponding sides of these
similar triangles? (Example: ΔABC and ΔA'B'C'?)
L
Similar by AA
Exercises: Use the completed chart from the Exploratory Challenge. For each
triangle, label the sides opp/adj/hyp. Use the terms opp/adj/hyp in setting up
proportions. Approximate the unknown lengths to one decimal place.
1. By what criteria is the triangle below similar to ΔDEF and ΔD'E'F' from your
completed chart? ________. Since it is similar, what will be the ratio of opp/
hyp? _______ What will be the ratio of adj/hyp?___________. Let's set up
proportions to solve for the missing sides:
2.
28
3.
4. From a point 120 m away from a building, Serena measures the angle between
0
the ground and the top of a building and finds it measures 41 .
What is the height of the building, to the nearest meter?
The table contains the values of the ratios opp/hyp and adj/hyp for right
0
0
0
0
triangles with acute angles {10 , 20 , 30 ,..., 90 }. Use the table to complete
each problem.
0
5. Find the approximate length, to one decimal place, of the leg opposite 80 .
6. Find the length of the hypotenuse to the nearest tenth.
700
7. Three city streets form a right triangle. Main Street and State Street are
0
perpendicular. Laura Street and State Street intersect at a 50 angle. The
distance along Laura Street to Main Street is 0.8 mile. If Laura Street is closed
between Main Street and State Street for a festival, approximately how far (to
the nearest tenth) will someone have to travel to get around the festival if they
take only Main Street and State Street?
Let's Sum it Up!
The hypotenuse of a right triangle is the side opposite the right angle; the other two sides
are called legs. For a given acute angle θ, the opposite side is the leg opposite θ; the
adjacent side is the leg that is contained in one of the two rays of that angle.
When two right triangles have equal corresponding acute angles, the triangles are similar,
and the ratios opp/hyp and adj/hyp are equal.
Name_____________________
CC Geometry H
Date _____________________
HW #22
#1-3 Use the completed chart from today's Exploratory Challenge.
Label the sides opp/adj/hyp appropriately. Use the terms opp/adj/hyp in setting
up proportions. Approximate the unknown lengths to one decimal place.
10
2.
1.
3.
410
6
x
x
67
0
z
28
z
y
0
22.5
y
#4-5 Use the completed chart to find missing lengths.
4. Find the length of the hypotenuse to the
nearest tenth.
50
0
5. Find the length of the leg adjacent to the 40 angle to the nearest hundredth
6. A cable anchors a utility pole to the ground as shown in the picture. The cable
0
forms an angle of 70 with the ground. The distance from the base of the utility
pole to the anchor point on the ground is 3.8 meters. Approximately how long, to
the nearest meter, is the support cable?
Review:
1. Find the values of the variables. Leave answers in simplest radical form, if
z
necessary.
3
2
y
x
2. Find the length of the missing sides of each triangle in simplest radical form, if
necessary.
a)
b)
3. Write the perimeter in simplest radical form:
c)