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
