Question

instructions
solve the following problem and select your answer from the choices given.

question

the area of the triangle above is 21. what is the value of x?

3
6
7
11

Explanation:

Step1: Write the formula for the area of a triangle

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\). Here, the base is \(x + 1\) and the height is \(x\), and \(A = 21\). So, \(21=\frac{1}{2}\times(x)(x + 1)\).

Step2: Simplify the equation

Multiply both sides of the equation by 2 to get \(42=x(x + 1)\). Expand the right - hand side: \(42=x^{2}+x\). Rearrange to get the quadratic equation \(x^{2}+x-42 = 0\).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \(-42\) and add up to \(1\). The numbers are \(7\) and \(-6\). So, \((x + 7)(x-6)=0\).

Step4: Solve for \(x\)

Set each factor equal to zero: \(x+7 = 0\) gives \(x=-7\), and \(x - 6=0\) gives \(x = 6\). Since \(x\) represents a length (height), \(x>0\).

Answer:

\(6\)