Download Mod 2 - Aim #28 - Manhasset Public Schools

Aim #28: How do we use trigonometry to find area of a triangle?
CC Geometry H
Do Now: Determine the area for each triangle below, if possible. If it is not
possible to find the area with the provided information, describe what is needed in
order to determine the area.
Example:
Given the third side length of the last triangle as 7, determine the
area of the triangle. (altitude HJ has been drawn.)
H
a) If GJ = x, represent JI, in terms of x: _______
12
7
h
2
G
x
I
J
15
c) Solve for x using substitution.
d) Solve for h.
e) Find the area of ΔGHI.
b) Write two expressions for h :
Using trigonometry to derive a new formula for the Area of a Triangle
ΔABC has sides a, b, and c, and altitude h is drawn from vertex A; m≮C = θ.
A
b
c
h
θ
B
C
a
a) Write an equation for the area of ΔABC. __________________
b) Write an equation for sin θ _________________
c) Solve the equation you wrote in (b) for h. _________________
d) Use your result in (c) to re-write the area formula in (a). _________________
New Formula for Area of a Triangle
a
A = ab sin θ
θ
b
where θ is the included angle between sides a and b.
Exercises
1) Find the area of the triangle to the nearest tenth.
2) A farmer is planning how to divide his land for planting next year‛s crops. A
triangular plot of land is left with two known lengths of 500 m and 1700 m, and
o
the angle between the two known side lengths is 30 . How could the farmer find
the area of the plot of land?
3) A real estate developer and her surveyor are searching for their next piece of
land to build on. They each examine a plot of land in the shape of ΔABC. The real
estate developer measures the length of AB and AC and finds them both to be
o
approximately 4000 ft., and the included angle has a measure of 50 . The surveyor
measures the length of AC and BC and finds the lengths to be approximately 4000
and 3400 feet and measures the angle between the two sides to be approximately
o
65 .
a) Draw two diagrams that model the situation, labeling all lengths and angle
measures
b) The real estate developer and surveyor each calculate the area of the plot of
land. Show how each person calculated the area, to the nearest foot.
c) Although each calculates roughly the same area, what might explain the
difference between the real estate agent and surveyor's calculated results?
4) Find the area of isosceles ΔPQR, with base QR = 11 and base angles with
0
measures of 71.45 .
5) Find the area, rounded to the nearest tenth.
6) A right rectangular pyramid has a square base with sides of length 5. Each
lateral face of the pyramid is an isosceles triangle. The angle on each lateral face
o
between the base of the triangle and the adjacent edge is 75 . Find the total
surface area of the pyramid, to the nearest tenth.
5
750
5
a lateral face
750
Let's Sum it Up!
For a triangle with side lengths a and b and included angle of measure θ,
the area formula is: Area of a Triangle =
absinθ
We need this formula when we are determining the area of a triangle and are not
provided with a height.
Name_________________________
Date: ___________________
CC Geometry H
HW #28
#1-3 Find the area of each triangle to the nearest tenth.
1)
2)
3)
o
4) In ΔDEF, EF = 15, DF = 20 and m≮F = 63 . Draw a labeled diagram and find the
area.
OVER
5) A landscape designer is designing a flower garden for a triangular area that is
bounded on two sides by the client‛s house and driveway. The length of the edges
of the garden along the house and driveway are 18 ft and 8 ft. respectively, and
the edges come together at an angle of 80 degrees. Draw a diagram, and then find
the area of the garden to the nearest square foot.
6) A regular hexagon is inscribed in a circle with a radius of 7. Find the perimeter
and area of the hexagon. [Hint: Divide the hexagon into equilateral triangles.]
7
Mixed Review: 1) Solve for the missing lengths, to the nearest hundredth.
a)
b)
64
12
0
c)
10
470
18
x
780
x
X
2) Solve for the indicated variables:
a)
10
y
450
b)
y
√15
x
x
o
3) The vertex angle of an isosceles triangle is 120 . The length of its base is 24 cm.
Find the altitude and find the length of each leg of the isosceles triangle.