Question

What is the expression for the base of the triangle?

The base of a triangle is four less than three times the height. The area of the triangle is 38 square centimeters.

Options:

A. 3h - 4 = 1/2(38 x h)

B. 4 - 3h

C. 3h - 4

D. 4h - 3

Answer 1

Which equation would be fit for the area of the triangle, substituting for the base?

Answer 2

To express the base of the triangle in terms of the height, we can start with the information provided:

1. The base \( b \) of the triangle is four less than three times the height \( h \).
This can be expressed mathematically as:
\[
b = 3h - 4
\]

2. The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
We know the area is 38 square centimeters, so we can set up the equation:
\[
38 = \frac{1}{2} \times b \times h
\]

3. Now, substituting for \( b \) in terms of \( h \) from the first equation into the area equation:
\[
38 = \frac{1}{2} \times (3h - 4) \times h
\]

From this context, we see the expression for the base of the triangle is simply:

\[
b = 3h - 4
\]

Looking at the options given:

A. \( 3h - 4 = \frac{1}{2}(38 \times h) \) \\
B. \( 4 - 3h \) \\
C. \( 3h - 4 \) \\
D. \( 4h - 3 \)

The correct choice that corresponds to the expression for the base \( b \) is:

**C. 3h - 4**

Answer 3

To find the correct equation for the area of the triangle while substituting for the base, we'll use the known formula for the area of a triangle and replace the base with the expression we derived earlier.

We know:
- The area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
- The base \( b \) is defined as:
\[
b = 3h - 4
\]
- The area is given as 38 square centimeters.

Now, substituting the expression for the base into the area equation gives us:
\[
38 = \frac{1}{2} \times (3h - 4) \times h
\]

To eliminate the fraction, we can multiply both sides of the equation by 2:
\[
76 = (3h - 4) \times h
\]

Now, distributing \( h \) on the right-hand side:
\[
76 = 3h^2 - 4h
\]

So, the equation that fits for the area of the triangle, substituting for the base, is:
\[
3h^2 - 4h - 76 = 0
\]

This is a quadratic equation that represents the relationship of height \( h \) in relation to the area given the base expression.