Question

the figure shows a pair of similar triangles. find the lengths of the sides labeled with the variables. △abc ~ △rst
answer
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st =
ac =

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle ABC\sim\triangle RST\), the ratios of corresponding sides are equal. That is \(\frac{AC}{RT}=\frac{BC}{ST}\). Substituting the given values, we get \(\frac{y - 1}{21}=\frac{2}{y}\).

Step2: Cross - multiply

Cross - multiplying the equation \(\frac{y - 1}{21}=\frac{2}{y}\) gives \(y(y - 1)=2\times21\). Expand the left - hand side: \(y^{2}-y = 42\).

Step3: Rearrange to a quadratic equation

Rearrange the equation \(y^{2}-y = 42\) to the standard quadratic form \(y^{2}-y - 42=0\).

Step4: Factor the quadratic equation

Factor \(y^{2}-y - 42\): \(y^{2}-y - 42=(y - 7)(y+6)=0\).

Step5: Solve for \(y\)

Set each factor equal to zero: \(y - 7 = 0\) or \(y+6 = 0\). Since the length of a side cannot be negative (\(y>0\)), we discard \(y=-6\). So \(y = 7\).

Step6: Find \(AC\)

Substitute \(y = 7\) into \(AC=y - 1\). Then \(AC=7 - 1=6\).

Answer:

\(ST = 7\)
\(AC = 6\)