Question

geometry with statistics honors - ferraro - 2 - a
topic 4: readiness assessment
what is the length of side a?
a. 169
b. 43
c. 13
d. 12

Explanation:

Step1: Consider right - triangle BCD

In right - triangle BCD, we know the height \(CD = 5\) and \(BD\) can be found. Since the perpendicular from \(C\) to \(AB\) divides \(AB\) into two parts and assume \(AD=x\), \(BD = 24 - x\). In right - triangle \(BCD\), we use the Pythagorean theorem \(a=\sqrt{CD^{2}+BD^{2}}\). First, we need to find \(BD\).
Let's consider the angle at \(C\). The perpendicular \(CD\) divides \(\angle C\) into two right - angled parts. We know that in right - triangle \(BCD\), we can find \(BD\) using trigonometry or the Pythagorean theorem directly. Since we have a right - triangle with height \(CD = 5\) and assume the base of the right - triangle \(BCD\) is \(BD\). Let's assume the right - triangle \(BCD\) has sides \(a\) (the hypotenuse we want to find), \(CD = 5\) and \(BD\).
We know that, by the Pythagorean theorem \(a=\sqrt{5^{2}+12^{2}}\) (assuming \(BD = 12\)). Here we assume that the perpendicular \(CD\) divides the base \(AB = 24\) symmetrically (a reasonable assumption for an isosceles - like triangle formed by the perpendicular from the vertex angle). In right - triangle \(BCD\), where \(CD = 5\) and \(BD=\frac{24}{2}=12\) (if the triangle is symmetric about the perpendicular \(CD\)).

Step2: Apply Pythagorean theorem

According to the Pythagorean theorem \(a=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Answer:

C. 13