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?

Explanation:

Step1: Recall the area formula for a triangle

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\). Let's assume the base of the large triangle (combined two smaller triangles) is \(b\) and the height is \(h=(x + 1+x)=2x + 1\). But we can also think of the two smaller triangles. If we assume the base of each smaller triangle is the same (since they share the same vertex - to - base structure), and let the base of each smaller triangle be \(b_1\) and \(b_2\) such that \(b_1 + b_2=b\). The area of the first smaller triangle \(A_1=\frac{1}{2}\times b_1\times x\), and the area of the second smaller triangle \(A_2=\frac{1}{2}\times b_2\times(x + 1)\). The total area \(A=A_1+A_2=\frac{1}{2}b_1x+\frac{1}{2}b_2(x + 1)=\frac{1}{2}(b_1x+b_2x + b_2)=\frac{1}{2}x(b_1 + b_2)+\frac{1}{2}b_2\). Another way: If we assume the base of the large triangle is \(b\) and we split the height into two parts \(x\) and \(x + 1\). The area \(A=\frac{1}{2}\times b\times(x+(x + 1))\). But a simpler approach: assume the base of the triangle is \(b\) (constant for both sub - triangles). The area of the first sub - triangle \(A_1=\frac{1}{2}bh_1\) with \(h_1=x\), and the area of the second sub - triangle \(A_2=\frac{1}{2}bh_2\) with \(h_2=x + 1\). Since \(A = A_1+A_2\) and \(A = 21\), we have \(21=\frac{1}{2}b(x+(x + 1))\). If we assume \(b = 6\) (by looking at the answer choices and testing, because if we use the formula \(A=\frac{1}{2}\times base\times height\) and assume the base is a whole number). Then \(21=\frac{1}{2}\times6\times(2x + 1)\).

Step2: Solve the equation

First, simplify the equation \(21=\frac{1}{2}\times6\times(2x + 1)\).

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Answer:

\(3\)