Perimeter and Area | A

Question 1

Deep Dive: In races, usually there is a common finish line for all the runners. Here are two square running tracks with the inner track of $100\text{ m}$ each side and outer track of $150\text{ m}$ each side. The common finishing line for both runners is shown by the flags in the figure which are in the center of one of the sides of the tracks.

If the total race is of $350\text{ m}$ , then we have to find out where the starting positions of the two runners should be on these two tracks so that they both have a common finishing line after they run for $350\text{ m}$ . Mark the starting points of the runner on the inner track as 'A' and the runner on the outer track as 'B'.

Solution

Understand the Question

Both runners need to cover a total distance of $350\text{ m}$ to reach a common finish line located at the midpoint of the bottom side of their respective tracks.
The inner square track has a side length of $100\text{ m}$ , and the outer square track has a side length of $150\text{ m}$ .
To determine each runner's starting position, we trace $350\text{ m}$ backwards from the finish line along each track.

Step 1 · Calculate Track Perimeters and Finish Line Positions

For the inner track (side $= 100\text{ m}$ ):

\begin{aligned} \text{Perimeter} &= 4 \times 100\text{ m} = 400\text{ m} \end{aligned}

Distance from the bottom corners to the finish line (midpoint):

\begin{aligned} 100\text{ m} \div 2 &= 50\text{ m} \end{aligned}

For the outer track (side $= 150\text{ m}$ ):

\begin{aligned} \text{Perimeter} &= 4 \times 150\text{ m} = 600\text{ m} \end{aligned}

Distance from the bottom corners to the finish line (midpoint):

\begin{aligned} 150\text{ m} \div 2 &= 75\text{ m} \end{aligned}

Step 2 · Find Starting Point A for the Inner Track

Runner A must run $350\text{ m}$ to reach the finish line. Tracing $350\text{ m}$ backwards from the finish line:

Finish line to bottom-left corner: $50\text{ m}$ (Remaining: $350\text{ m} - 50\text{ m} = 300\text{ m}$ )
Up along the left side: $100\text{ m}$ (Remaining: $300\text{ m} - 100\text{ m} = 200\text{ m}$ )
Across the top side: $100\text{ m}$ (Remaining: $200\text{ m} - 100\text{ m} = 100\text{ m}$ )
Down along the right side: $100\text{ m}$ (Remaining: $100\text{ m} - 100\text{ m} = 0\text{ m}$ )

Thus, starting point A is located at the bottom-right corner of the inner track.

$\text{Check: } 100\text{ m} + 100\text{ m} + 100\text{ m} + 50\text{ m} = 350\text{ m}$

Step 3 · Find Starting Point B for the Outer Track

Diagram 1

Since the perimeter of the outer track is $600\text{ m}$ , running $350\text{ m}$ to the finish line means starting point B is located $600\text{ m} - 350\text{ m} = 250\text{ m}$ forward from the finish line:

Finish line to bottom-right corner: $75\text{ m}$ (Remaining: $250\text{ m} - 75\text{ m} = 175\text{ m}$ )
Up along the right side: $150\text{ m}$ (Remaining: $175\text{ m} - 150\text{ m} = 25\text{ m}$ )
Left along the top side: $25\text{ m}$ (Remaining: $25\text{ m} - 25\text{ m} = 0\text{ m}$ )

Thus, starting point B is on the top side, $25\text{ m}$ from the top-right corner.

$\text{Check: } (150\text{ m} - 25\text{ m}) + 150\text{ m} + 75\text{ m} = 125\text{ m} + 150\text{ m} + 75\text{ m} = 350\text{ m}$

Answer

Point A (Inner track): At the bottom-right corner.

Point B (Outer track): On the top side, $25\text{ m}$ from the top-right corner.

Common Mistakes

Measuring From the Corner: Assuming the finish line is at a corner rather than at the center of the side ( $50\text{ m}$ and $75\text{ m}$ from the corners).
Direction Confusion: Tracing forward instead of backward when locating the starting point from the finish line.

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