Question

What are the values of a and b​

Answer 1

Answer:

$\large\boxed{B.\ a=8,\ b=4\sqrt5}$

Step-by-step explanation:

We have three similar triangles:

$\triangle ABC\sim\triangle DBA\sim\triangle DAC$

therefore the sides are in proportion:

$\dfrac{AD}{CD}=\dfrac{BD}{AD}$

We have AD = a, CD = 16 and BD = 4. Substitute:

$\dfrac{a}{16}=\dfrac{4}{a}$              cross multiply

$(a)(a)=(16)(4)\\\\a^2=64\to a=\sqrt{64}\\\\a=8$

and other proportion:

$\dfrac{AB}{BC}=\dfrac{BD}{AB}$

We have AB = b, BC = 16 + 4 = 20 and BD = 4. Substitute:

$\dfrac{b}{20}=\dfrac{4}{b}$             cross multiply

$(b)(b)=(20)(4)\\\\b^2=80\to b=\sqrt{80}\\\\b=\sqrt{16\cdot5}\\\\b=\sqrt{16}\cdot\sqrt{5}\\\\b=4\sqrt{5}$