Question

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

Explanation:

Step1: Find length of 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

\(\triangle ABC\sim\triangle CAD\) (by AA similarity since \(\angle B=\angle A = 90^{\circ}\) and \(\angle ACB+\angle BAC = 90^{\circ}\), \(\angle ACB+\angle ACD = 90^{\circ}\), so \(\angle BAC=\angle ACD\)).
Then \(\frac{AB}{BC}=\frac{AC}{CD}\).

Step3: Substitute values and solve for CD

We know \(AB = 5\), \(BC=\sqrt{11}\), \(AC = 6\). Substituting into \(\frac{AB}{BC}=\frac{AC}{CD}\), we get \(CD=\frac{AC\times BC}{AB}=\frac{6\times\sqrt{11}}{5}\approx\frac{6\times3.317}{5}=\frac{19.902}{5}= 3.9804
eq\) any of the given options. There is a mistake above.

Let's use another way.
Since \(\triangle ABC\sim\triangle CAD\), we have \(\frac{AB}{AC}=\frac{AC}{CD}\).

Step1: Set up proportion

We know \(AB = 5\) and \(AC = 6\). The proportion \(\frac{AB}{AC}=\frac{AC}{CD}\) gives us \(CD=\frac{AC^{2}}{AB}\).

Step2: Calculate CD

Substitute \(AB = 5\) and \(AC = 6\) into the formula \(CD=\frac{6^{2}}{5}=\frac{36}{5}=7.2\).

Answer:

\(7.2\)