Question

as shown in the figure below, in right triangle abc, m∠bac = 90°, ad ⊥ bc at point d, if bd : cd = 3 : 1, and ab = 6, then the length of ac is _______. a 2 b 3 c 2√3 d 3√2

Explanation:

Step1: Use similarity of triangles

In right - triangle $ABC$ with $\angle BAC = 90^{\circ}$ and $AD\perp BC$, we have $\triangle ABD\sim\triangle CAD\sim\triangle CBA$. From the similarity of $\triangle ABD$ and $\triangle CAD$, we know that $\frac{BD}{AD}=\frac{AD}{CD}$, that is $AD^{2}=BD\times CD$. Let $CD = x$, then $BD = 3x$. Also, from the similarity of $\triangle ABD$ and $\triangle CBA$, we have $\frac{AB}{BC}=\frac{BD}{AB}$, and from the similarity of $\triangle CAD$ and $\triangle CBA$, we have $\frac{AC}{BC}=\frac{CD}{AC}$.

Step2: Apply the property of right - triangle and similarity

We know that in right - triangle $ABC$, by the Pythagorean theorem $AB^{2}+AC^{2}=BC^{2}=(BD + CD)^{2}=(3x+x)^{2}=16x^{2}$. Also, from the similarity $\frac{AB^{2}}{AC^{2}}=\frac{BD}{CD}=\frac{3x}{x}=3$.

Step3: Solve for $AC$

Since $\frac{AB^{2}}{AC^{2}} = 3$ and $AB = 6$, then $\frac{36}{AC^{2}}=3$. Cross - multiply to get $3AC^{2}=36$, then $AC^{2}=12$, and $AC = 2\sqrt{3}$.

Answer:

C. $2\sqrt{3}$