Area And Perimeter In The Coordinate Plane Problems Independent ... - Free Printable

Problem Analysis:

The image contains two problems related to the coordinate plane. Let's solve each problem step by step.

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Problem 1: What is the area of triangle \( \triangle ABC \)?

#### Step 1: Identify the coordinates of points \( A \), \( B \), and \( C \).
From the graph:
- Point \( A \) is at \( (0, 0) \).
- Point \( B \) is at \( (6, 0) \).
- Point \( C \) is at \( (0, 4) \).

#### Step 2: Use the formula for the area of a triangle.
The area \( A \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[
A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substitute the coordinates of \( A(0, 0) \), \( B(6, 0) \), and \( C(0, 4) \):
\[
A = \frac{1}{2} \left| 0(0 - 4) + 6(4 - 0) + 0(0 - 0) \right|
\]
Simplify:
\[
A = \frac{1}{2} \left| 0 + 24 + 0 \right| = \frac{1}{2} \times 24 = 12
\]

#### Step 3: Verify using the base and height method.
The base \( AB \) is along the x-axis from \( (0, 0) \) to \( (6, 0) \), so the length of the base is:
\[
\text{Base} = 6
\]
The height is the vertical distance from \( C(0, 4) \) to the x-axis, which is:
\[
\text{Height} = 4
\]
The area of the triangle is:
\[
A = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 6 \times 4 = 12
\]

#### Final Answer for Problem 1:
\[
\boxed{12}
\]

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Problem 2: What are the area and perimeter of rectangle \( MNOQ \)?

#### Step 1: Identify the coordinates of points \( M \), \( N \), \( O \), and \( Q \).
From the graph:
- Point \( M \) is at \( (1, 1) \).
- Point \( N \) is at \( (5, 1) \).
- Point \( O \) is at \( (1, 5) \).
- Point \( Q \) is at \( (5, 5) \).

#### Step 2: Calculate the length and width of the rectangle.
- The length \( MN \) is the horizontal distance between \( M(1, 1) \) and \( N(5, 1) \):
\[
\text{Length} = 5 - 1 = 4
\]
- The width \( MO \) is the vertical distance between \( M(1, 1) \) and \( O(1, 5) \):
\[
\text{Width} = 5 - 1 = 4
\]

#### Step 3: Calculate the area of the rectangle.
The area \( A \) of a rectangle is given by:
\[
A = \text{Length} \times \text{Width} = 4 \times 4 = 16
\]

#### Step 4: Calculate the perimeter of the rectangle.
The perimeter \( P \) of a rectangle is given by:
\[
P = 2 \times (\text{Length} + \text{Width}) = 2 \times (4 + 4) = 2 \times 8 = 16
\]

#### Final Answers for Problem 2:
\[
\text{Area} = 16, \quad \text{Perimeter} = 16
\]

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Final Answers:

1. Area of \( \triangle ABC \): \(\boxed{12}\)
2. Area of rectangle \( MNOQ \): \(\boxed{16}\), Perimeter of rectangle \( MNOQ \): \(\boxed{16}\)