Question

sp# 9 9. if △gfe ~ △ghj, find the value of x.

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle GFE\sim\triangle GHJ\), the ratios of corresponding sides are equal. So, \(\frac{EF}{HJ}=\frac{EG}{HG}\).

Step2: Substitute the given values

We have \(EF = x + 6\), \(HJ=7.5\), \(EG = 12\), and \(HG=2x + 1\). Then \(\frac{x + 6}{7.5}=\frac{12}{2x + 1}\).

Step3: Cross - multiply

Cross - multiplying gives \((x + 6)(2x + 1)=12\times7.5\).

Expand the left - hand side: \(2x^{2}+x+12x + 6=90\).

Simplify to get \(2x^{2}+13x+6 - 90=0\), i.e., \(2x^{2}+13x - 84=0\).

Step4: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 2\), \(b = 13\), \(c=-84\)), 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 length cannot be negative, we reject \(x=-10.5\).

Answer:

\(x = 4\)