Question

look at the figure below: what is the length, in units, of segment cd? 11 7.2 5.5 10

Explanation:

Step1: Find BC in right - triangle ABC

In right - triangle ABC with AB = 5 and AC = 6, by the Pythagorean theorem \(BC=\sqrt{AC^{2}-AB^{2}}\). So \(BC=\sqrt{6^{2}-5^{2}}=\sqrt{36 - 25}=\sqrt{11}\).

Step2: Prove similarity of triangles

Triangles ABC and CAD are similar. Since \(\angle B=\angle A = 90^{\circ}\) and \(\angle ACB+\angle BAC = 90^{\circ}\), \(\angle BAC+\angle CAD=90^{\circ}\), so \(\angle ACB=\angle CAD\). Then \(\triangle ABC\sim\triangle CAD\).

Step3: Set up proportion for similar triangles

The ratio of corresponding sides of similar triangles is equal. We have \(\frac{AB}{BC}=\frac{BC}{CD}\). Substituting AB = 5 and \(BC = \sqrt{11}\), we get \(5:\sqrt{11}=\sqrt{11}:CD\). Cross - multiplying gives \(5\times CD=11\), so \(CD = 7.2\).

Answer:

7.2