Question

The graph below shows the locations of a triangular park and a rectangular park with all coordinates shown measured in kilometers.

Triangular park A-B-C has its vertices A at (-10, -6), B at (-6, -6) and C at (-1, 6). Rectangular park D-E-F-G has its vertices D at (0, 2), E at (4, -1), F at (10, 7) and G at (6, 10).

Use the drop-down arrows to choose the response that makes each statement true.

The triangular park has a perimeter of Query 1
kilometers and an area of Query 2
square kilometers.
The rectangular park has a perimeter of Query 3
kilometers and an area of Query 4
square kilometers.

Answer 1

To calculate the perimeter and area for both the triangular park (A-B-C) and the rectangular park (D-E-F-G), we can follow these steps:

### Triangular Park A-B-C:

**1. Perimeter Calculation:**

First, we need to calculate the lengths of the sides of the triangle.

- Length AB:
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-6 - (-10))^2 + (-6 - (-6))^2} = \sqrt{(4)^2 + (0)^2} = 4
\]

- Length BC:
\[
BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(-1 - (-6))^2 + (6 - (-6))^2} = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]

- Length CA:
\[
CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2} = \sqrt{(-10 - (-1))^2 + (-6 - (6))^2} = \sqrt{(-9)^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15
\]

Now we can find the perimeter:
\[
\text{Perimeter} = AB + BC + CA = 4 + 13 + 15 = 32 \text{ kilometers}
\]

**2. Area Calculation:**

To find the area of triangle A-B-C, we can use the formula for the area of a triangle given its vertices:
\[
\text{Area} = \frac{1}{2} \left| x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B) \right|
\]
Substituting in the coordinates:
\[
\text{Area} = \frac{1}{2} \left| -10(-6 - 6) + -6(6 - (-6)) + -1(-6 - (-6)) \right|
\]
\[
= \frac{1}{2} \left| -10(-12) + -6(12) + -1(0) \right|
\]
\[
= \frac{1}{2} \left| 120 - 72 + 0 \right| = \frac{1}{2} \times 48 = 24 \text{ square kilometers}
\]

### Rectangular Park D-E-F-G:

**1. Perimeter Calculation:**

For the rectangle, we find the lengths of adjacent sides (DE and EF):

- Length DE:
\[
DE = \sqrt{(4 - 0)^2 + (-1 - 2)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

- Length EF:
\[
EF = \sqrt{(10 - 4)^2 + (7 - (-1))^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]

The perimeter \( P \) of a rectangle is given by:
\[
P = 2 \times (length + width) = 2 \times (5 + 10) = 2 \times 15 = 30 \text{ kilometers}
\]

**2. Area Calculation:**

The area \( A \) of a rectangle is given by:
\[
A = length \times width = 5 \times 10 = 50 \text{ square kilometers}
\]

### Summary of Results:
- Perimeter of triangular park: **32 kilometers**
- Area of triangular park: **24 square kilometers**
- Perimeter of rectangular park: **30 kilometers**
- Area of rectangular park: **50 square kilometers**

So, we fill in the responses as follows:

The triangular park has a perimeter of **32** kilometers and an area of **24** square kilometers.
The rectangular park has a perimeter of **30** kilometers and an area of **50** square kilometers.