Question

sp# 9 9. if △gfe ~ △ghj, find the value of x.
3p# 12~ 10. △xzw ~ △xyv, find the perimeter of △xzw.

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle GFE\sim\triangle GHJ\), the ratios of corresponding sides are equal. That is \(\frac{GF}{GH}=\frac{GE}{GJ}\). Substituting the given values, we get \(\frac{x + 6}{7.5}=\frac{12}{2x+1}\).

Step2: Cross - multiply

Cross - multiplying the equation \(\frac{x + 6}{7.5}=\frac{12}{2x+1}\) gives \((x + 6)(2x+1)=12\times7.5\).
Expanding the left - hand side: \(2x^{2}+x+12x + 6=90\), which simplifies to \(2x^{2}+13x+6 - 90=0\), or \(2x^{2}+13x - 84=0\).

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 2\), \(b = 13\), \(c=-84\)), we can use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(13)^{2}-4\times2\times(-84)=169 + 672=841\).
Then \(x=\frac{-13\pm\sqrt{841}}{4}=\frac{-13\pm29}{4}\).
We have two solutions: \(x_{1}=\frac{-13 + 29}{4}=\frac{16}{4}=4\) and \(x_{2}=\frac{-13-29}{4}=\frac{-42}{4}=-10.5\).
Since \(x\) represents a length, we discard the negative solution. So \(x = 4\).

Answer:

\(x = 4\)