The base of a triangular piece of fabric is 8in more than the height. The area is 600in2. Find the base and height of the triangle. Round your answer to the nearest tenth of a inch.

To solve for the base and height of the triangle, we'll use the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Given:
- The area of the triangle \( \text{Area} = 600 \, \text{in}^2 \)
- The base is 8 inches more than the height. If we let the height be \( h \) inches, then the base will be \( h + 8 \) inches.

Using the area formula:

\[
600 = \frac{1}{2} \times (h + 8) \times h
\]

Simplify and solve for \( h \):

\[
600 = \frac{1}{2} \times (h^2 + 8h)
\]

Multiply both sides by 2 to eliminate the fraction:

\[
1200 = h^2 + 8h
\]

Now, rearrange the equation into a standard quadratic form:

\[
h^2 + 8h - 1200 = 0
\]

This quadratic equation can be solved using the quadratic formula:

\[
h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 8 \), and \( c = -1200 \). Substituting these values into the formula:

\[
h = \frac{-8 \pm \sqrt{8^2 - 4(1)(-1200)}}{2(1)}
\]

\[
h = \frac{-8 \pm \sqrt{64 + 4800}}{2}
\]

\[
h = \frac{-8 \pm \sqrt{4864}}{2}
\]

\[
h = \frac{-8 \pm 69.7}{2}
\]

Now, calculate the two possible values for \( h \):

1. \( h = \frac{-8 + 69.7}{2} = \frac{61.7}{2} = 30.85 \)
2. \( h = \frac{-8 - 69.7}{2} = \frac{-77.7}{2} = -38.85 \)

Since height cannot be negative, the valid height is:

\[
h \approx 30.85 \, \text{inches}
\]

Now, calculate the base:

\[
\text{base} = h + 8 \approx 30.85 + 8 = 38.85 \, \text{inches}
\]

So, the base is approximately \( 38.9 \) inches, and the height is approximately \( 30.9 \) inches.

Would you like further details or have any questions? Here are five related questions:

1. How do you derive the quadratic formula?
2. What is the geometric significance of a triangle's area?
3. How would the problem change if the area were different?
4. What are the properties of a triangle with a specific base and height ratio?
5. How would you verify the solution using an alternative method?

**Tip:** When solving quadratic equations, always check for both positive and negative roots, and ensure the solution makes sense in the context of the problem.

Learn how to find the base and height of a triangle given its area of 600 square inches. Follow a detailed step-by-step solution using the area formula and quadratic equation to determine both dimensions accurately. Suitable for Grades 9-12 students.

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