Question

look at the figure below: what is the length, in units, of segment cd? 5.50 12.25 8 14

Explanation:

Step1: Find \(BC\) using Pythagorean theorem

In right - triangle \(ABC\), by Pythagorean theorem \(BC=\sqrt{AC^{2}-AB^{2}}\). Given \(AB = 4\) and \(AC = 7\), then \(BC=\sqrt{7^{2}-4^{2}}=\sqrt{49 - 16}=\sqrt{33}\).

Step2: Prove \(\triangle ABC\sim\triangle CAD\)

Since \(\angle B=\angle CAD = 90^{\circ}\) and \(\angle ACB+\angle BAC=90^{\circ}\), \(\angle BAC+\angle CAD = 90^{\circ}\), so \(\angle ACB=\angle CAD\). By AA (angle - angle) similarity criterion, \(\triangle ABC\sim\triangle CAD\).

Step3: Use similarity ratio

For similar triangles \(\triangle ABC\) and \(\triangle CAD\), the ratio \(\frac{AB}{AC}=\frac{AC}{CD}\). Substituting \(AB = 4\) and \(AC = 7\) into \(\frac{4}{7}=\frac{7}{CD}\). Cross - multiply gives \(4CD=49\), then \(CD=\frac{49}{4}=12.25\).

Answer:

12.25