Chapter 7 Coordinate Geometry

Case Study 1: Urban Planning and Zoning

Case Description:
An urban planning team is designing a new rectangular park in a city. The park will be placed on a coordinate plane, with its vertices labeled $A(2, 3)$, $B(2, 8)$, $C(10, 8)$, and $D(10, 3)$. The planners are considering dividing the park into two sections: a playground and a garden. The division will be made by drawing a line segment from $A$ to $C$. The team needs to calculate the length of the diagonal, verify the park’s shape as a rectangle by examining its properties, and determine the midpoint of the diagonal to mark the division point.

MCQs:

1. What is the length of the diagonal $AC$ of the rectangular park?

   • A) 6 units
   • B) 8 units
   • C) 10 units
   • D) 12 units

2. What is the distance between points $B$ and $C$?

   • A) 8 units
   • B) 6 units
   • C) 4 units
   • D) 5 units

3. What are the coordinates of the midpoint of $AC$, which will serve as the division point?

   • A) (5, 5.5)
   • B) (6, 5)
   • C) (6, 4)
   • D) (8, 6)

4. Which statement is correct regarding the shape of $ABCD$?

   • A) It is a square.
   • B) It is a trapezium.
   • C) It is a rectangle.
   • D) It is a parallelogram.

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Case Study 2: Treasure Hunt on an Island

Case Description:
A treasure hunt event is planned on a rectangular island. The coordinates of the corners of the island are $P(3, 4)$, $Q(3, 12)$, $R(15, 12)$, and $S(15, 4)$. The event organizers have buried a treasure at the point equidistant from points $Q$ and $S$. Participants are given clues to locate this point using the coordinate geometry concepts of midpoint and distance formula. They need to calculate the distances from various points and determine the exact location of the treasure.

MCQs:

1. What is the distance between $P$ and $S$?

   • A) 12 units
   • B) 10 units
   • C) 16 units
   • D) 14 units

2. What are the coordinates of the midpoint of $QS$, where the treasure is buried?

   • A) (9, 8)
   • B) (7, 6)
   • C) (8, 10)
   • D) (10, 8)

3. What is the distance between $Q$ and $S$?

   • A) 14 units
   • B) 12 units
   • C) 16 units
   • D) 18 units

4. Which other point, besides $P$, is also equidistant from both $Q$ and $S$?

   • A) Point R
   • B) Point S
   • C) The midpoint of PR
   • D) The midpoint of PS

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Case Study 3: Warehouse Layout Design

Case Description:
A company plans to design a rectangular warehouse with diagonal paths for easy movement between corners. The vertices of the warehouse are located at $A(1, 1)$, $B(1, 9)$, $C(13, 9)$, and $D(13, 1)$ on a coordinate grid. To improve efficiency, the company wants to construct paths along the diagonals from $A$ to $C$ and $B$ to $D$. The designers need to determine the lengths of these diagonals, confirm the shape of the warehouse as a rectangle, and identify the coordinates of their intersection point.

MCQs:

1. What is the length of the diagonal $AC$?

   • A) 10 units
   • B) 12 units
   • C) 15 units
   • D) 13 units

2. What is the distance between points $A$ and $B$?

   • A) 6 units
   • B) 8 units
   • C) 9 units
   • D) 10 units

3. What are the coordinates of the intersection point of the diagonals $AC$ and $BD$?

   • A) (7, 5)
   • B) (6, 4)
   • C) (7, 7)
   • D) (8, 5)

4. Which shape best describes the warehouse based on the coordinates of the vertices?

   • A) Square
   • B) Parallelogram
   • C) Rectangle
   • D) Trapezium

Case Study 4: A Triangular Plot of Land

Case Description:
A farmer owns a triangular plot of land, with vertices marked at $M(4, 1)$, $N(10, 1)$, and $O(4, 8)$. He wants to divide the land equally by connecting the midpoint of side $MO$ to vertex $N$. This midpoint will serve as a point of reference for further farm planning, including irrigation and crop placement. He needs to calculate the length of each side, locate the midpoint, and confirm that this line effectively divides the plot into two equal areas.

MCQs:

1. What is the length of side $MN$?

   • A) 6 units
   • B) 8 units
   • C) 10 units
   • D) 12 units

2. What are the coordinates of the midpoint of side $MO$?

   • A) (4, 4.5)
   • B) (5, 5)
   • C) (3, 4.5)
   • D) (4, 5)

3. What is the distance between points $M$ and $O$?

   • A) 6 units
   • B) 7 units
   • C) 8 units
   • D) 10 units

4. Which of the following statements is correct about the line connecting the midpoint of $MO$ to $N$?

   • A) It divides the triangle into two congruent triangles.
   • B) It is perpendicular to $NO$.
   • C) It is parallel to $MO$.
   • D) It forms an isosceles triangle with $MN$.

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Case Study 5: Mapping a Triangular Garden

Case Description:
A triangular garden is mapped with vertices at points $X(2, 3)$, $Y(8, 3)$, and $Z(5, 7)$. The garden will have flower beds arranged along the medians. The median from $X$ to side $YZ$ is of particular interest for the garden’s main pathway. To prepare for planting, the gardener needs to determine the lengths of each side, locate the midpoint of $YZ$, and measure the length of the median to aid in the pathway design.

MCQs:

1. What is the length of side $XY$?

   • A) 5 units
   • B) 6 units
   • C) 8 units
   • D) 10 units

2. What are the coordinates of the midpoint of side $YZ$?

   • A) (6.5, 5)
   • B) (5, 5)
   • C) (7, 4)
   • D) (3, 6)

3. What is the length of the median from $X$ to $YZ$?

   • A) 5 units
   • B) 6 units
   • C) 7 units
   • D) 8 units

4. Which statement is true about the median from $X$ to $YZ$?

   • A) It bisects $YZ$ at right angles.
   • B) It divides the triangle into two triangles of equal area.
   • C) It is equal in length to $XY$.
   • D) It runs parallel to $YZ$.