Question

finding lengths of the sides of a rectangle
a fitness club wants to set up stations for four different workouts
around the shelves of free weights in the middle of the room. the
trainer wants the distance from the free weights to each station to be
the same, so he uses a rectangular shape, as shown in the diagram.
what is the distance from the free weights to the push - up station?
distance from the jump - rope station to the sit - up station?
what is the distance from the push - up station to the jump - rope station?
what is the distance from the lunge station to the jump - rope station?

Explanation:

Step1: Recall properties of a rectangle

In a rectangle, the diagonals are equal and bisect each other.

Step2: Find distance from free - weights to push - up station

Since the diagonals bisect each other, the distance from free weights to push - up station is equal to the distance from free weights to sit - up station. So it is \(5\) ft (because if one half of a diagonal is \(2.5\) ft, the full diagonal is \(2\times2.5 = 5\) ft and the diagonals of a rectangle are equal, so the distance from free weights to push - up station (half of the other diagonal) is also \(5\) ft).

Step3: Find distance from jump - rope station to sit - up station

In a rectangle, opposite sides are equal. The side adjacent to the \(3\) ft side (from push - ups to lunges) and the side we want (jump - rope to sit - ups) are opposite. Using the Pythagorean theorem for the right - triangle formed by the sides of the rectangle (\(a = 3\) ft, \(b\) (jump - rope to sit - ups), \(c\) (diagonal \(= 5\) ft)). By \(a^{2}+b^{2}=c^{2}\), \(b=\sqrt{c^{2}-a^{2}}=\sqrt{5^{2}-3^{2}}=\sqrt{25 - 9}=\sqrt{16}=4\) ft. Wait, no, actually, in a rectangle, opposite sides are equal. The side from jump - rope to sit - ups is equal to the side from push - ups to lunges. Wait, no, another approach: The diagonals of a rectangle bisect each other. The distance from free weights to each station is the same. The distance from push - up to jump - rope: Using the Pythagorean theorem in the right - triangle formed by the sides of the rectangle. If one side is \(3\) ft (push - up to lunge) and the other side (lunge to sit - up) we found the diagonal related. Wait, no, the distance from push - up to jump - rope: The diagonals of the rectangle. Wait, no, the formula for the length of the diagonal of a rectangle with sides \(l\) and \(w\) is \(d=\sqrt{l^{2}+w^{2}}\). But we know from the property of rectangle diagonals bisecting each other. The distance from free weights to each station. The distance from push - up to jump - rope: Since the diagonals of a rectangle are equal and bisect each other. If half of one diagonal is \(2.5\) ft (free weights to sit - up) and half of the other diagonal (free weights to push - up) is \(2.5\) ft (wait no, no, the diagonals are equal. The length of the diagonal of the rectangle: if we consider the right - triangle with sides \(3\) ft and \(4\) ft (from Pythagorean triple \(3 - 4-5\)), but wait, the distance from push - up to jump - rope is the length of the diagonal of the rectangle. Using the Pythagorean theorem: if one side is \(3\) ft (push - up to lunge) and the other side (lunge to sit - up) we can find. Wait, no, the distance from push - up to jump - rope: Let's use the property of rectangle diagonals. The diagonals of a rectangle are equal. If we consider the two right - triangles formed by the diagonal. The length of the diagonal \(d\): we know that from the half - diagonal. Wait, the distance from free weights to each station. The diagonals of the rectangle are equal. So the length of the diagonal (push - up to jump - rope) is \(5\) ft (because if half of a diagonal is \(2.5\) ft, full diagonal is \(5\) ft).

Step4: Find distance from lunge station to jump - rope station

Using the Pythagorean theorem in the right - triangle formed by the sides of the rectangle. One side is \(3\) ft (push - up to lunge) and the other side (lunge to sit - up) we found using the diagonal. Wait, no, in a rectangle, opposite sides are equal. The distance from lunge to jump - rope is equal to the distance from push - up to sit - up. Using the Pythagorean theorem: if the diagonal is \(5\) ft (push - up to jump - rope) and one side is \(3\) ft (push - up to lunge), then the other side (lunge to jump - rope) is \(4\) ft (\(a = 3\), \(c = 5\), \(b=\sqrt{5^{2}-3^{2}}=4\))

Answer:

• Distance from free weights to push - up station: \(5\) ft
• Distance from jump - rope station to sit - up station: \(4\) ft
• Distance from push - up station to jump - rope station: \(5\) ft
• Distance from lunge station to jump - rope station: \(4\) ft