Aim #9: How do we find perimeter and area in the Cartesian plane? CC Geometry H
Do Now: Find the area of the shaded region. Leave answer in terms of π when
necessary.
a)
b)
4
3
1
2
6
­1
­4
­3
2
­2
­2
­3
­4
1) Plot the points O (0,0), A (5,2) and B (3,4)
a) Find the perimeter of ΔOAB in simplest radical
form and to the nearest hundredth.
b) Find the area of ΔOAB using decomposition. Start by enclosing the triangle
with a rectangle.
c) When is it easier apply the triangle area formula and not use decomposition?
3
4
Can we develop a general formula for the area of any triangle with one vertex at
(0,0) using decomposition?
Let's call O (0,0), A (x1, y1) and B (x2, y2)
a) What is the area of the rectangle?
b) What is the area of the surrounding triangles?
Left:
Right:
Bottom:
c) Write the general formula to find the area of triangle OAB:
d) Expand the formula and simplify it:
This formula works for any triangle with one vertex at (0,0), no matter the
quadrant, and the area will be positive as long as (x1, y1) is the next point
in a counterclockwise direction.
Find the area of the triangles with vertices listed by using the formula:
b) O(0,0)
A (5,-3)
B (-2,6)
a) O (0,0)
A (4,1)
B (5,6)
The general formula we found for a triangle with one vertex at the origin can be
applied to any triangle on the Cartesian plane. A triangle with coordinates:
A(x1, y1), B(x2, y2) and C(x3,y3) would apply the formula:
This is called the "shoelace" area formula. Here is how to remember it:
x1y1
x2y2
x3y3
x1y1
Given the triangle below with vertices A(-2,3), B(-1,-2) and C(4,3), find the area
using the shoelace formula. Confirm the area found by using the formula for area
of a triangle
A
C
B
Given the triangle, find the area using the shoelace formula.
Find the value of q so that the area of the triangle determined by the points
A(-3, q), B(5, 2), and C(3, 7) is 12 square units.
Let's Sum it Up!
To find the perimeter of a triangle, use the distance formula or Pythagorean
Theorem to find the length of each segment and add the distances together.
To find the area of a triangle:
-you can use the triangle area formula
-you can use the decomposition method.
-if one vertex is at the origin, you can apply the formula:
-or you can use the shoelace formula:
Name____________________
CC Geometry H
Date __________
HW #9
1) Compute the perimeter and area of each polygon. When necessary, round
to the nearest hundredth.
a)
b)
2) Given the figures below, find the area by decomposing into rectangles and
triangles. You can decompose within a figure also!
a)
c)
b)
d)
3) Given the triangles below, find the area using the shoelace formula.
a)
b)
Review
1) Find the value of x. State the theorem used to find x.
9x+16
19x+3
6x+15
2) A regular pentagon is shown to the right:
If the pentagon is rotated clockwise
around its center, the minimum number
of degrees it must be rotated to carry
the pentagon onto itself is: _________.