Question

the triangles formed on the line are similar. using the slope of the line, how could you determine the value of a? select all that apply.
options:

• $\frac{a}{2} = \frac{1}{10}; a = \frac{1}{5}$
• $\frac{2}{10} = \frac{a}{1}; a = \frac{1}{5}$
• $\frac{a}{1} = \frac{10}{2}; a = 5$
• $\frac{a}{10} = \frac{1}{2}; a = 5$

Explanation:

Step1: Identify slope ratio for red triangle

Slope = $\frac{10}{2} = 5$

Step2: Identify slope ratio for green triangle

Slope = $\frac{a}{1}$

Step3: Set ratios equal (similar triangles)

$\frac{a}{1} = \frac{10}{2}$ → $a = 5$

Step4: Check valid equations

$\frac{a}{10} = \frac{1}{2}$ → $a=5$; $\frac{1}{a} = \frac{2}{10}$ → $a=5$ are correct. $\frac{a}{2} = \frac{1}{10}$ gives $a=\frac{1}{5}$ (incorrect), but if the green triangle’s horizontal side is 1 and vertical is a, the correct ratio is $\frac{a}{1}=\frac{10}{2}$ leading to $a=5$. The valid options are those resulting in $a=5$.

Answer:

$\square \frac{a}{2} = \frac{1}{10}; a = \frac{1}{5}$
$\square \frac{a}{10} = \frac{1}{2}; a = 5$
$\square \frac{1}{a} = \frac{2}{10}; a = 5$
$\square \frac{a}{10} = \frac{1}{2}; a = 5$