Answer:
62.5 feet
Explanation:
The markings on the angles in the figure tell you the triangles are similar. Similar triangles have proportional corresponding sides. This is used to write and solve an equation for x.
Similarity
The angles in each triangle are marked with the following symbols:
- square corner - signifying a right angle
- one arc
- two arcs
The purpose of the marks is to signify that angles with the same mark are congruent. When the angles of one triangle are congruent to the angles of another triangle, the two triangles are similar.
Similar triangles have another feature: corresponding sides are proportional. That is, their ratios are identical.
This can be taken a couple of different ways:
- sides of one triangle have the same ratio to each other as the corresponding sides of the other triangle
- corresponding sides of the two triangles have the same ratio
Application
In the given triangles, we can identify the corresponding sides as ...
smaller triangle : larger triangle
between right angle and double arc angle — 36 ft : 50 ft
hypotenuse — 45 ft : x ft
Our description of similarity tells us we can write the statements of proportionality as either of ...
- 36 : 50 = 45 : x . . . . . side ratios in the same triangle are the same
- 36 : 45 = 50 : x . . . . . side ratios between triangles are the same
Solving the proportion
For proportions written in this way, the "outer" and "inner" products are the same. This is sometimes expressed by saying "the product of the means is equal to the product of the extremes."
a : b = c : d ⇒ ad = bc
When the ratios are written as fractions, this result is what you get from "cross multiplication":
a/b = c/d ⇒ ad = bc . . . . . the result of multiplying both sides by bd
Once the proportion is written in this product form, the value of any of the variables can be found by dividing both sides of the equation by the coefficient of that variable.
Desired distance
Using either of the proportions we wrote above, the product form is ...
36x = 45·50
x = (45·50)/36 . . . . . . divide by the coefficient of x
x ≈ 62.5 . . . . feet
The distance from the playground to the swimming pool is 62.5 feet.