Land in downtown Columbia is valued at $20 a square foot. What is the value of a triangular lot with sides of lengths 112, 146, and 186 ft? (Round your answer to the nearest dollar.)

To find the value of the triangular lot, we first need to calculate its area and then multiply it by the value per square foot.

### Step 1: Use Heron's formula to find the area of the triangle.
Heron's formula is used to find the area of a triangle when the lengths of all three sides are known. The formula is:

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]

where:
- $A$ is the area,
- $a$, $b$, and $c$ are the lengths of the sides of the triangle,
- $s$ is the semi-perimeter, calculated as:

\[
s = \frac{a + b + c}{2}
\]

### Step 2: Calculate the semi-perimeter.
For the given triangle with side lengths $a = 112$, $b = 146$, and $c = 186$:

\[
s = \frac{112 + 146 + 186}{2} = \frac{444}{2} = 222
\]

### Step 3: Apply Heron's formula.
Substitute the values of $s$, $a$, $b$, and $c$ into the formula:

\[
A = \sqrt{222(222 - 112)(222 - 146)(222 - 186)}
\]
\[
A = \sqrt{222(110)(76)(36)}
\]

Now, let's calculate this.

The area of the triangular lot is approximately 8,174 square feet.

### Step 4: Calculate the value of the lot.
The value of the land is $20$ dollars per square foot. Therefore, the total value of the lot is:

\[
\text{Value} = 20 \times 8174 = 163,480 \text{ dollars}
\]

Thus, the value of the triangular lot is approximately **$163,480**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does Heron's formula apply to other types of triangles?
2. How would the problem change if the lot were valued per acre instead of square feet?
3. Can you calculate the perimeter of this triangular lot?
4. What if the land value changes? How would it affect the total price?
5. How do you calculate the area of irregularly shaped lots?

**Tip**: Heron's formula works for any triangle, not just right-angled ones, making it versatile for real-world applications.

Land in downtown Columbia is valued at $20 a square foot. What is the value of a triangular lot with sides of lengths 112, 146, and 186 ft? (Round your answer to the nearest dollar.)

Learn how to calculate the value of a triangular lot with side lengths of 112 ft, 146 ft, and 186 ft using Heron's formula. This geometry problem helps you find the area of the triangle and its value based on $20 per square foot. Suitable for high school students in Grades 9-12.

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