The coordinate grid shows an equilateral triangle that fits inside a square.

GM0203_PRT_035A-72dpi.png

a. Find the area of the square. Show your work.

s =
, so A =
2 =
units2

b. Find the area of the triangle. Show your work.

b =
, h=3⎯⎯√,
so A=12(
)(3⎯⎯√)=3⎯⎯√
units2.

Part B
Select the correct choices to complete the sentence.
c. Find the area of the square that is not covered by the triangle. Select an exact value and then round to the nearest tenth. Justify your reasoning.

The area not covered by the triangle is 1 of 5.
Select Choice
the area of the square 2 of 5.
Select Choice
the area of the triangle. So, A 3 of 5.
Select Choice
(
16 4 of 5.
Select Choice
3⎯⎯√)
units2 or about 5 of 5.
Select Choice
units2.

To solve this problem, we will go step by step for each part. Here are the details:

### Part a: Find the area of the square

To find the area of the square, we need to know the length of its side \( s \). Since the equilateral triangle fits perfectly inside the square, and all sides of the square are equal, we assume the triangle's base aligns with one side of the square.

Let's assume \( s \) is the side length of the square.

- **Area of the square:** 
  \[
  A = s^2
  \]

So, 
\[
s = \text{(side length of the square)},
\]
\[
A = s^2 \text{ square units}
\]

### Part b: Find the area of the triangle

For an equilateral triangle, the area can be found using the formula:
\[
A = \frac{1}{2} \times b \times h
\]
where \( b \) is the base of the triangle, and \( h \) is the height.

Given that \( h = \sqrt{3} \) (which is the height of the equilateral triangle when the side length \( b = s \)), the area of the triangle is:
\[
A = \frac{1}{2} \times b \times \sqrt{3}
\]
Since \( b = s \), we substitute and get:
\[
A = \frac{1}{2} \times s \times \sqrt{3} = \frac{s \sqrt{3}}{2} \text{ square units}
\]

### Part c: Find the area of the square that is not covered by the triangle

To find the area of the square that is not covered by the triangle, we subtract the area of the triangle from the area of the square.

1. **The area of the square** is \( s^2 \) square units.
2. **The area of the triangle** is \( \frac{s \sqrt{3}}{2} \) square units.

The area not covered by the triangle is:
\[
A_{\text{not covered}} = s^2 - \frac{s \sqrt{3}}{2} \text{ square units}
\]

**Exact Value:**
\[
A_{\text{not covered}} = s^2 - \frac{s \sqrt{3}}{2} 
\]

**Rounded to the nearest tenth:**
- This depends on the specific value of \( s \). If you have the value of \( s \), you can calculate the exact area not covered and round it accordingly.

### Filling in the blanks for Part C:
- **The area not covered by the triangle is:**  
  \(\frac{s^2 - \frac{s \sqrt{3}}{2}}{s^2}\) or you could say in terms of the specific parts of the question.

To help you fill in the final blanks, I'll need the specific numerical value of \( s \) if it was provided, or I can assume it based on context if you want to provide that. 

Would you like the exact computation for any specific value of \( s \) or any further details on this?

Learn how to calculate the area of a square and an equilateral triangle that fits inside. Step-by-step solution with detailed formulas and explanations. Suitable for Grades 7-9.

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