Question

if line segment bc is considered the base of triangle abc, what is the corresponding height of the triangle?
0.625 units
0.8 units
1.25 units
1.6 units

Explanation:

Step1: Identify coordinates

Points: $A(-2,1)$, $B(3,2)$, $C(-2,-1)$

Step2: Calculate length of base $BC$

Use distance formula:
$\text{Length of } BC = \sqrt{(3 - (-2))^2 + (2 - (-1))^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$

Step3: Calculate area of $\triangle ABC$

Use horizontal base $AC$ (since $A$ and $C$ share $x$-coordinate):
$\text{Length of } AC = 1 - (-1) = 2$
Horizontal distance from $B$ to $AC$ is $3 - (-2) = 5$
$\text{Area} = \frac{1}{2} \times 2 \times 5 = 5$

Step4: Solve for height $h$

Use area formula with base $BC$:
$\text{Area} = \frac{1}{2} \times BC \times h$
$5 = \frac{1}{2} \times \sqrt{34} \times h$
$h = \frac{10}{\sqrt{34}} = \frac{10\sqrt{34}}{34} = \frac{5\sqrt{34}}{17} \approx 1.6$

Answer:

1.6 units