Coordinate Geometry (Case Study) (Class 10 Mathematics)

Real-world coordinate geometry problems involving distances, areas, and straight lines. Applications include map navigation, construction, and architecture

Coordinate Geometry (Case Study) (Class 10 Mathematics)

Real-world coordinate geometry problems involving distances, areas, and straight lines. Applications include map navigation, construction, and architecture

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TL;DR: Real-world coordinate geometry problems involving distances, areas, and straight lines. Applications include map navigation, construction, and archite…

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Real-world coordinate geometry problems involving distances, areas, and straight lines. Applications include map navigation, construction, and architecture

Coordinate Geometry (Case Study) MCQs with Answers & Explanations

Q1. A surveyor maps a triangular plot with vertices at A(0,0), B(6,0), and C(3,4). What is the area of this plot?

  1. 18 sq units
  2. 6 sq units
  3. 24 sq units
  4. 12 sq units ✓ (correct)

Explanation: Using the formula: Area = 0.5 * |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| = 0.5 * |0(0-4) + 6(4-0) + 3(0-0)| = 0.5 * 24 = 12 sq units.

Q2. Two towns are located at coordinates (0,0) and (8,6) on a map. What is the straight-line distance between them in units?

  1. 10 units ✓ (correct)
  2. 12 units
  3. 15 units
  4. 14 units

Explanation: Distance = sqrt((8-0)^2 + (6-0)^2) = sqrt(64+36) = sqrt(100) = 10 units.

Q3. A line passes through points (1,2) and (5,10). What is the slope of this line?

  1. 1
  2. 2 ✓ (correct)
  3. 4
  4. 3

Explanation: Slope = (y2-y1)/(x2-x1) = (10-2)/(5-1) = 8/4 = 2.

Q4. If a line has slope 2 and passes through the point (3,7), what is the equation of the line in the form y = mx + c?

  1. y = 2x + 3
  2. y = 2x + 7
  3. y = 2x + 1 ✓ (correct)
  4. y = 3x + 1

Explanation: Using y = mx + c with m=2: 7 = 2(3) + c, so c = 1. Equation: y = 2x + 1.

Q5. Points P(1,1), Q(4,4), and R(7,7) are collinear. This means what?

  1. They form an equilateral triangle
  2. They form a right angle
  3. They are at equal distances from origin
  4. They lie on the same straight line ✓ (correct)

Explanation: Collinear means the points lie on the same straight line. Slope PQ = (4-1)/(4-1) = 1. Slope QR = (7-4)/(7-4) = 1. Equal slopes confirm collinearity.

Q6. A rectangle has vertices at (0,0), (5,0), (5,3), and (0,3). What is its perimeter?

  1. 16 units ✓ (correct)
  2. 8 units
  3. 15 units
  4. 20 units

Explanation: Length = 5, Width = 3. Perimeter = 2(5+3) = 16 units.

Q7. The midpoint of a line segment joining (2,4) and (6,8) is?

  1. (5,7)
  2. (4,6) ✓ (correct)
  3. (3,5)
  4. (4,5)

Explanation: Midpoint = ((2+6)/2, (4+8)/2) = (4, 6).

Q8. Which of the following lines is parallel to y = 3x + 2?

  1. y = -3x + 2
  2. y = 2x + 3
  3. y = 3x - 5 ✓ (correct)
  4. y = (1/3)x + 2

Explanation: Parallel lines have the same slope. y = 3x - 5 has slope 3, same as y = 3x + 2. All other options have different slopes.

Frequently Asked Questions

How do we find the equation of a line given two points?

First, calculate the slope m = (y2-y1)/(x2-x1). Then use the point-slope form y - y1 = m(x - x1) with either point, or solve for the y-intercept c using y = mx + c. Alternatively, use the two-point form: (y - y1)/(y2 - y1) = (x - x1)/(x2 - x1), which directly gives the line equation.

What is the significance of the y-intercept in the equation y = mx + c?

The y-intercept (value c) represents the point where the line crosses the y-axis, at coordinates (0, c). It shows the value of y when x = 0. The slope m tells us how steeply the line rises or falls, and the y-intercept gives us a reference point to graph the line.

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