Question

1. determine the value of x. be aware of the order of the triangles vertices!!!!

△gfe ~ △gqr

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle GFE\sim\triangle GQR\), the ratios of corresponding sides are equal. That is \(\frac{GF}{GQ}=\frac{GE}{GR}\). Here \(GF = 11x + 1\), \(GQ=(11x + 1)+6\), \(GE = 30\), \(GR = 9\).
So, \(\frac{11x + 1}{(11x + 1)+6}=\frac{30}{9}\). Cross - multiply gives \(9(11x + 1)=30(11x + 1+6)\).

Step2: Expand the equation

Expand \(9(11x + 1)=99x+9\) and \(30(11x + 7)=330x+210\).
The equation becomes \(99x + 9=330x+210\).

Step3: Solve for \(x\)

Subtract \(99x\) from both sides: \(9 = 231x+210\).
Subtract \(210\) from both sides: \(231x=9 - 210=-201\).
Then \(x=\frac{-201}{231}=-\frac{67}{77}\). Wait, there is a mistake. Let's use the correct ratio \(\frac{GF}{GQ}=\frac{GE}{GQ}\) (corrected ratio: since \(\triangle GFE\sim\triangle GQR\), \(\frac{GF}{GQ}=\frac{GE}{GQ}\) is wrong. The correct ratio is \(\frac{GF}{GQ}=\frac{GE}{GQ}\) (no, correct is \(\frac{GF}{GR}=\frac{GE}{GQ}\)).

Correct Step1:
Since \(\triangle GFE\sim\triangle GQR\), \(\frac{GF}{GR}=\frac{GE}{GQ}\). So \(\frac{11x + 1}{6}=\frac{30}{9}\).
Cross - multiply: \(9(11x + 1)=30\times6\).

Step2:

Expand \(99x+9 = 180\).
Subtract \(9\) from both sides: \(99x=180 - 9=171\).
Divide both sides by \(99\): \(x=\frac{171}{99}=\frac{19}{11}\approx1.73\).

Another way:
Since \(\triangle GFE\sim\triangle GQR\), the ratio of corresponding sides.
\(\frac{GF}{GQ}=\frac{GE}{GR}\) (no, \(\frac{GF}{GR}=\frac{GE}{GQ}\)).
\(\frac{11x + 1}{6}=\frac{30}{9}\)
\(9(11x + 1)=180\)
\(99x+9 = 180\)
\(99x=171\)
\(x=\frac{171}{99}=\frac{19}{11}\)

Answer:

\(x = \frac{19}{11}\)