Question

In triangle ABC, the length of the base AC is represented by (2x+1 ), and the height of the triangle is represented by (x+3). What is the area of triangle ABC expressed as a trinomial in terms of x?

Answer 1

To find the area \( A \) of triangle ABC, we use the formula for the area of a triangle:

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

In this case, the base AC is given as \( 2x + 1 \) and the height is given as \( x + 3 \). Plugging these expressions into the formula, we get:

\[
A = \frac{1}{2} \times (2x + 1) \times (x + 3)
\]

To simplify this expression, we first multiply the two expressions in the parentheses:

\[
(2x + 1)(x + 3)
\]

Using the distributive property (also known as the FOIL method), we compute:

\[
(2x + 1)(x + 3) = 2x \cdot x + 2x \cdot 3 + 1 \cdot x + 1 \cdot 3
\]

Calculating each term, we have:

\[
2x^2 + 6x + x + 3 = 2x^2 + 7x + 3
\]

Now substituting this back into the area formula:

\[
A = \frac{1}{2} \times (2x^2 + 7x + 3)
\]

Next, we distribute \( \frac{1}{2} \):

\[
A = \frac{1}{2} \cdot 2x^2 + \frac{1}{2} \cdot 7x + \frac{1}{2} \cdot 3 = x^2 + \frac{7}{2}x + \frac{3}{2}
\]

Thus, the area of triangle ABC expressed as a trinomial in terms of \( x \) is:

\[
\boxed{x^2 + \frac{7}{2}x + \frac{3}{2}}
\]