A new park, Park At The Falls, is being created near your neighborhood. The head of the HOA has commissioned you
to build a geometrically shaped sandbox for the park. You sketch a drawing of the sandbox and head to the store to
purchase the lumber to serve as the edging of the sandbox. How many linear feet of lumber do you need?

The small sandbox (triangle ABC) needs 39 linear feet of lumber and the big sandbox (triangle XYZ) needs 52 feet of lumber.

Procedure - Determination of the lumber needed for the edging of a sandbox

In this question we shall use the principle of similarity between triangles. According to the image attached we know that [tex]\triangle ABC \sim \triangle XYZ[/tex], which means that the following relationship is observed:

[tex]\frac{AB}{XY} = \frac{AC}{XZ} = \frac{BC}{ZY}[/tex] (1)

Please notice that all side lengths are measured in linear feet.

If we know that [tex]AB = 18[/tex], [tex]XY = 24[/tex], [tex]XZ = 16[/tex] and [tex]ZY = 12[/tex], then the lengths of the sides [tex]AC[/tex] and [tex]BC[/tex] are:

[tex]AC = \left(\frac{XZ}{XY} \right)\cdot AB[/tex]

[tex]AC = \left(\frac{16}{24} \right)\cdot (18)[/tex]

[tex]AC = 12[/tex]

[tex]BC = \left(\frac{ZY}{XY} \right)\cdot AB[/tex]

[tex]BC = \left(\frac{12}{24} \right)\cdot (18)[/tex]

[tex]BC = 9[/tex]

Determination of the perimeter of each sandbox

The perimeter of a triangle is the sum of the lengths of the triangle. We proceed to calculate the perimeter of each triangle:

Triangle ABC

[tex]p_{ABC} = 39\,ft[/tex] [tex]\blacksquare[/tex]

Triangle XYZ

[tex]p_{XYZ} = 52\,ft[/tex] [tex]\blacksquare[/tex]

The small sandbox (triangle ABC) needs 39 linear feet of lumber and the big sandbox (triangle XYZ) needs 52 feet of lumber. [tex]\blacksquare[/tex]

To learn more on similar triangles, we kindly invite to check this verified question: https://brainly.com/question/25882965