Question

each unit on the coordinate grid represents 1 yard. the rectangular pool and triangular hot tub shown are both in need of covers. how much total material is needed to cover both? square yards

Explanation:

Step1: Find rectangle side lengths

Use distance formula for vertical side: from (0,4) to (3,2):
$\text{Length}_1 = \sqrt{(3-0)^2 + (2-4)^2} = \sqrt{9+4} = \sqrt{13}$
Use distance formula for horizontal side: from (0,4) to (4,10):
$\text{Length}_2 = \sqrt{(4-0)^2 + (10-4)^2} = \sqrt{16+36} = \sqrt{52} = 2\sqrt{13}$

Step2: Calculate rectangle area

Area = $\text{Length}_1 \times \text{Length}_2$
$Area_{rect} = \sqrt{13} \times 2\sqrt{13} = 2 \times 13 = 26$

Step3: Find triangle base/height

Base = 2 yards, Height = 3 yards

Step4: Calculate triangle area

$Area_{tri} = \frac{1}{2} \times \text{base} \times \text{height}$
$Area_{tri} = \frac{1}{2} \times 2 \times 3 = 3$

Step5: Sum the two areas

Total area = $26 + 3$

Answer:

29 square yards