https://file.aiquickdraw.com/mathbot/file/1727316964495vc82gl6c.jpg

Let's solve the problem in the image, which deals with a right triangle window.

We are given:
- The height of the window is represented by the expression \(6x - 7\) feet.
- The area of the window is represented by the expression \(24x - 28\) square feet.

Since the window is in the shape of a right triangle, we can use the formula for the area of a right triangle, which is:

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

Let the base of the triangle be \(b\). Then, using the area formula:

\[
\frac{1}{2} \times b \times (6x - 7) = 24x - 28
\]

To eliminate the fraction, multiply both sides by 2:

\[
b \times (6x - 7) = 2 \times (24x - 28)
\]

Simplify the right-hand side:

\[
b \times (6x - 7) = 48x - 56
\]

Now, solve for \(b\) by dividing both sides by \(6x - 7\):

\[
b = \frac{48x - 56}{6x - 7}
\]

Notice that the numerator and the denominator can both be factored:

\[
b = \frac{8(6x - 7)}{6x - 7}
\]

Cancel out the common factor \(6x - 7\):

\[
b = 8
\]

So, the base of the window is 8 feet.

Would you like more details on any part of the solution? Here are 5 related questions:

1. What if the base of the triangle was given, and you needed to find the height instead?
2. How do you factor more complex polynomials?
3. How would this problem change if the shape of the window was a rectangle?
4. What does it mean to cancel out factors in an equation?
5. How would the problem change if the window was an equilateral triangle?

**Tip**: Always check whether terms can be factored or simplified to make the calculation easier.

Tricia is building the framework for an entryway window for a house under construction. The window is in the shape of a right triangle. The height of the window is represented by the expression 6x - 7 feet. The area of the window is represented by the expression 24x - 28 square feet. What is the base of the window in feet?

Solve for the base of a right triangle where the height is given by 6x - 7 feet and the area by 24x - 28 square feet. This problem involves algebraic factoring and geometric concepts. Suitable for students in Grades 8-10.

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