Question

question 17 of 25
what is the length of $overline{ac}$?
a. 136
b. 72
c. 132
d. 96

Explanation:

Step1: Use similar - triangles property

Since \(\triangle BAC\sim\triangle CED\) (by AA similarity, as \(\angle BAC=\angle CED = 90^{\circ}\) and \(\angle ABC+\angle ACB = 90^{\circ}\), \(\angle ECD+\angle ACB = 90^{\circ}\), so \(\angle ABC=\angle ECD\)), we have the proportion \(\frac{BA}{CE}=\frac{AC}{ED}\). Substituting the given values, we get \(\frac{51}{x}=\frac{144 - x}{3}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{51}{x}=\frac{144 - x}{3}\) gives us \(51\times3=x(144 - x)\). So, \(153 = 144x-x^{2}\), which can be rewritten as \(x^{2}-144x + 153=0\).

Step3: Solve for \(x\)

We can also use the fact that from the proportion \(\frac{51}{x}=\frac{144 - x}{3}\), cross - multiplying gives \(153=144x - x^{2}\), or \(x^{2}-144x + 153 = 0\). Another way is to note that if we assume the two right - triangles are similar, we can use the ratio of their sides. Let's use the property of similar triangles directly. Since \(\frac{51}{x}=\frac{144 - x}{3}\), we get \(153=144x - x^{2}\). Rearranging to \(x^{2}-144x + 153 = 0\). However, we can also use the fact that \(\frac{51}{3}=\frac{144 - x}{x}\), then \(51x = 3(144 - x)\), \(51x=432 - 3x\), \(51x+3x=432\), \(54x = 432\), \(x = 8\).

Step4: Calculate \(AC\)

\(AC=144 - x\), substituting \(x = 8\) into the equation, we get \(AC=144 - 8=136\).

Answer:

A. 136