NAME _________________________________
COMMON CORE GEOMETRY
Module 4 Part II
Perimeters and Areas of Polygonal Regions in the Cartesian Plane
DATE
4/9
PAGE
2-3
4/10
4-5
4/13
4-5
4/14
6-7
4/15
4/16
8-10
8-10
4/17
TOPIC
Lesson 1: Perimeter and Area of Triangles in the
plane
Lesson 2: Perimeter and Area of Polygons in the
coordinate plane
More practice Lesson 1 and 2
Quiz
Lesson 3: Perimeters and Areas of Polygonal regions
defined by systems of inequalities
Lesson 4: Dividing Segments Proportionally
Lesson 4 Continued
Quiz
QUEST
1
HOMEWORK
Lesson #1 worksheet
Lesson #2 worksheet
Lesson #3 worksheet
Lesson #4 worksheet
Study
Lesson 1: Perimeter and Area of Triangles in the Cartesian Plane
Opening Exercises
Find the area of the shaded regions.
a.
b.
Example 1
Given the segment pictured to the right, answer the following:
a) What is the change in y?
b) What is the change in x?
c) What is the slope?
d) Find the length of the segment in simplest
radical form.
Example 2
Given the triangle below with vertices A(‐2, 3), B(4, 4) and C(-1, ‐2).
a. Calculate the exact perimeter of ∆ABC .
b. Calculate the area of ∆ABC .
2
Example 3
Consider a triangular region in the plane with vertices 𝑂(0, 0), 𝐴(5, 2), and 𝐵(3, 4).
a. What is the perimeter of the triangular region?
b. Find the area of the triangle with vertices listed, by finding the area of the rectangle enclosing the triangle
and subtracting the area of the surrounding triangles.
Example 4
Find the area of the triangle with vertices listed, by finding the area of the rectangle enclosing the triangle and
subtracting the area of the surrounding triangles?
𝑂(0, 0), 𝐴(3, 2), 𝐵 (−2, 6)
3
Lesson 2: Perimeter and Area of Polygonal Regions in the Cartesian Plane
Opening Exercise
Find the area of the triangle given. Compare your answer and method to your neighbor’s and discuss
differences.
Exercises 1–6
1.
Given rectangle 𝐴𝐵𝐶𝐷:
a. Identify the vertices.
b. Find the perimeter.
c. Find the area by boxing in the rectangle and subtracting off the extra.
d. Find the area using the area formula.
4
2.
Calculate the area and perimeter of the given quadrilateral.
3.
Find the area and perimeter of the hexagon shown.
5
Lesson 3: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
Opening Exercise
Graph the following:
1
2
a. 𝑦 < 2 𝑥 − 4
b. 𝑦 ≥ − 3 𝑥 + 5
Exercises 1–2
1.
A quadrilateral region is defined by the system of inequalities below:
𝑦 ≤𝑥+6
𝑦 ≤ −2𝑥 + 12
𝑦 ≥ 2𝑥 − 4
a. Sketch the region.
b. Determine the vertices of the quadrilateral.
c.
Find the perimeter of the quadrilateral region.
d. Find the area of the quadrilateral region.
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𝑦 ≥ −𝑥 + 2
2.
A quadrilateral region is defined by the system of inequalities below:
𝑦 ≤𝑥+5
𝑦 ≥𝑥−4
𝑦≤4
a. Sketch the region.
b. Determine the vertices of the quadrilateral.
c.
What type of quadrilateral is defined by these inequalities?
Justify your answer.
d. Find the perimeter of the quadrilateral region.
e. Find the area of the quadrilateral region.
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5
𝑦 ≥ −4𝑥 − 4
Lesson 4: Dividing Segments Proportionately
Exercises 1–3
1.
̅̅̅ given 𝑆(−2, 8) and 𝑇(10, −4).
Find the midpoint of ̅𝑆𝑇
2. Graph the points P(2,0) and Q(18,16).
a. Divide the segment into two equal
pieces.
b. Divide the segment into 4th’s
c. Divide the segment into 8th’s
d. What are the coordinates of the point that is 3/8ths the distance from P to Q?
e. What are the coordinates of the point that is 7/8ths the distance from P to Q?
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3. Find the point on the directed segment from (−5, −3) to (4,9) that divides it in the ratio of 1: 3.
Discussion
Given the pictured line segment, locate the value of the point that is:
a. 1/ 2 of the way from 10 to 20
b. 3/ 4 of the way from 10 to 20
c. 2/5 of the way from 10 to 20
Example 1
Using the problems above, think about the work that is needed to answer each question:
How did we determine the number of units that are in each partition?
How did we find the distance from the starting point to the new point?
In general how do we find the actual location of our answer?
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Example 2
Using your knowledge from the previous example, locate the point that is 2/3 of the way from 201 to 312.
Example 3
Find the point on the directed segment from (−2,0) to (5,8) that divides it in the ratio of 1: 3.
Example 4
Find the point on the directed segment from (−𝟑, −𝟐) to (𝟒, 𝟖) that divides it into a ratio of 𝟑: 𝟐
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