Question

3 solve for x in the given figures
a) given \\( \triangle a b d \sim \triangle c b d \\)
b) given \\( \triangle a y z \sim \triangle s y z \\)

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle ABD\sim\triangle CBD\), the ratios of corresponding sides are equal. That is \(\frac{AB}{CB}=\frac{BD}{BD}=\frac{AD}{CD}\). But more straightforwardly, for similar triangles \(\triangle ABD\) and \(\triangle CBD\), we can use the proportion \(\frac{AD}{BD}=\frac{BD}{CD}\).

Step2: Substitute the known values

We know \(BD = 12\), \(CD=16\). Let \(AD=x\). Then the proportion is \(\frac{x}{12}=\frac{12}{16}\).

Step3: Cross - multiply to solve for \(x\)

Cross - multiplying gives \(16x=12\times12\). So \(16x = 144\). Then \(x=\frac{144}{16}=9\).

Answer:

\(x = 9\)