Question

if \\( \triangle snd\sim\triangle lwd \\), find the value of \\( x \\).

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle SND\sim\triangle LWD\), the ratios of corresponding sides are equal. That is \(\frac{SN}{LW}=\frac{SD}{LD}\).

We know \(SD = 30\), \(SL=12\), so \(LD=SD - SL=30 - 12 = 18\). And \(SN = 6x + 1\), \(LW=2x + 7\).

So the proportion is \(\frac{6x + 1}{2x+7}=\frac{30}{18}\).

Step2: Simplify the proportion

Simplify \(\frac{30}{18}=\frac{5}{3}\). Then our equation becomes \(\frac{6x + 1}{2x+7}=\frac{5}{3}\).

Cross - multiply: \(3(6x + 1)=5(2x + 7)\).

Expand both sides: \(18x+3 = 10x + 35\).

Step3: Solve for \(x\)

Subtract \(10x\) from both sides: \(18x-10x+3=10x - 10x+35\), which gives \(8x+3 = 35\).

Subtract \(3\) from both sides: \(8x+3 - 3=35 - 3\), so \(8x=32\).

Divide both sides by \(8\): \(x=\frac{32}{8}=4\).

Answer:

\(4\)