Case Study Class 10 Coordinate Geometry

Quadratic Equations Case Study Class 10

Case Study Class 10 Coordinate Geometry

Understanding Coordinate Geometry Case Study Questions Class 10 is essential for students preparing for board exams. These Case Study Questions Class 10 allow learners to apply the concepts of distance formula, section formula, and area of triangles in real-life contexts. Since Class 10 is a milestone year, solving Math Case Study Class 10 helps students strengthen concepts and build exam confidence.

Importance of Case Study Class 10 Coordinate Geometry

Practicing Case Study Class 10 enhances logical thinking and analytical skills. Students learn to connect geometry formulas with real-world problems like navigation, architecture, and city planning. Teachers recommend Case Study Questions Class 10 for thorough revision as they highlight both strengths and weak areas. By solving such case studies, learners develop deeper understanding and problem-solving ability.

Preparation with Online Tests

Our free online practice tests on Case Study Questions Class 10 include step-by-step solutions and explanations. These Math Case Study Questions Class 10 cover a wide variety of problem situations to ensure effective learning. Regular practice with Case Study Class 10 questions boosts accuracy, confidence, and performance in CBSE exams.

Case Study 3: Virtual Reality Game

Case Study 3: Virtual Reality Game

A group of friends, Arjun, Bilal, and Charlie, are playing a virtual reality game where they have to navigate a maze. The game environment is a 2D coordinate plane. The starting points of their avatars are at $A(-2, -3)$, $B(4, 5)$, and $C(10, -3)$ respectively. They discover a hidden treasure chest at a point $P$. To unlock the treasure, they must calculate the area of the triangular region formed by their starting positions. They also need to find the location of a secret key that is located at the centroid of this triangular region. The game’s rules state that the path from Arjun’s avatar to Bilal’s avatar must be bisected by a checkpoint. They must also find the coordinates of a monster that is located at a point that divides the line segment connecting Bilal and Charlie in the ratio $1:3$. To make it more challenging, the game’s AI modifies the coordinates of Arjun’s avatar to $(x, y)$ such that the distance between Arjun and Charlie is $10$ units. They must find the possible coordinates for Arjun’s new position.

1. What is the area of the triangular region formed by the starting points of the three friends?

  • A) 48 sq. units
  • B) 54 sq. units
  • C) 60 sq. units
  • D) 72 sq. units
Answer: A) 48 sq. units
Solution: Using the area of a triangle formula with vertices $A(-2, -3)$, $B(4, 5)$, and $C(10, -3)$: \[ \text{Area} = \tfrac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right| \] \[ = \tfrac{1}{2} \left| (-2)(5 – (-3)) + 4(-3 – (-3)) + 10(-3 – 5) \right| \] \[ = \tfrac{1}{2} \left| (-2)(8) + 0 + 10(-8) \right| = \tfrac{1}{2} |-96| = 48 \] Hence, the area is $48$ sq. units.

2. Find the coordinates of the secret key, which is located at the centroid of $\triangle ABC$.

  • A) $(4, -\tfrac{1}{3})$
  • B) $(4, 1)$
  • C) $(5, -\tfrac{1}{3})$
  • D) $(5, 1)$
Answer: A) $(4, -\tfrac{1}{3})$
Solution: The centroid of a triangle is \[ G\left(\tfrac{x_1+x_2+x_3}{3}, \tfrac{y_1+y_2+y_3}{3}\right) \] Substituting: \[ \bar{x} = \tfrac{-2+4+10}{3} = 4, \quad \bar{y} = \tfrac{-3+5+(-3)}{3} = -\tfrac{1}{3} \] So, the centroid is $(4, -\tfrac{1}{3})$.

3. What are the coordinates of the checkpoint that bisects the path from Arjun to Bilal?

  • A) $(1, 1)$
  • B) $(1, 2)$
  • C) $(2, 1)$
  • D) $(2, 2)$
Answer: A) $(1, 1)$
Solution: Midpoint of $AB$: \[ M = \left(\tfrac{-2+4}{2}, \tfrac{-3+5}{2}\right) = (1,1) \]

4. Find the coordinates of the monster that divides $BC$ internally in the ratio $1:3$.

  • A) $(7, 3)$
  • B) $(5.5, 3)$
  • C) $(5.5, 4)$
  • D) $(7, 4)$
Answer: B) $(5.5, 3)$
Solution: Section formula: \[ x = \tfrac{1\cdot 10 + 3\cdot 4}{4} = 5.5, \quad y = \tfrac{1\cdot (-3) + 3\cdot 5}{4} = 3 \] So, coordinates are $(5.5, 3)$.

5. If Arjun’s new position is $(x,y)$ such that the distance from $C(10, -3)$ is $10$, which of the following could be his new position?

  • A) $(16, 5)$
  • B) $(4, 5)$
  • C) $(10, 7)$
  • D) $(12, 5)$
Answer: C) $(10, 7)$
Solution: Using distance formula: \[ (x-10)^2 + (y+3)^2 = 100 \] Check option (C): $(10,7)$ \[ (10-10)^2 + (7+3)^2 = 0+100 = 100 \quad \checkmark \] Hence, $(10,7)$ is a valid solution.

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