Question

question 7 of 10
what is the value of x in the figure below? in this diagram, $\triangle abd \sim \triangle cad$.
image of a right triangle with a perpendicular segment from the right angle to the hypotenuse, labeled with points c, d, b on the hypotenuse, cd and db labeled 16 (total hypotenuse length? wait, no, the diagram shows c to d to b, with cb = 16? wait, the diagram: c---d---b, with cd + db = 16? wait, the diagram has c to d to b, with d between c and b, ad perpendicular to cb, ab = 10, and we need to find x = db. the triangles abd and cad are similar.
\bigcirc a. $\frac{25}{4}$
\bigcirc b. 8
\bigcirc c. $\frac{16}{5}$
\bigcirc d. $\sqrt{5}$
\bigcirc e. 16
\bigcirc f. $\sqrt{160}$

Explanation:

Step1: Recall Similar Triangles Property

Since \(\triangle ABD \sim \triangle CAD\), the corresponding sides are proportional. So, \(\frac{BD}{AD}=\frac{AD}{CD}=\frac{AB}{CA}\). But for finding \(x = BD\), we use the proportion of corresponding sides: \(\frac{BD}{AB}=\frac{AB}{BC}\).

Step2: Identify Known Values

We know \(AB = 10\), \(BC = 16\), and \(BD = x\). Substituting into the proportion \(\frac{x}{10}=\frac{10}{16}\).

Step3: Solve for \(x\)

Cross - multiply: \(16x=10\times10 = 100\). Then \(x=\frac{100}{16}=\frac{25}{4}\).

Answer:

A. \(\frac{25}{4}\)