Chapter 1 Important Questions: Orienting Yourself: The Use of Coordinates

The coordinates of the origin are (0, 0). The origin is the point of intersection of the x-axis and y-axis.

If a point lies on the x-axis, its y-coordinate is 0. The coordinates of any point on the x-axis are of the form (x, 0).

The point P(−3, 5) lies in Quadrant II. In Quadrant II, the x-coordinate is negative and the y-coordinate is positive.

The x-axis is horizontal and the y-axis is vertical.

B) (x, y) = (y, x) only when x = y. Two ordered pairs are equal only when their corresponding coordinates are equal. If x ≠ y, then (x, y) ≠ (y, x).

To find distance AD:

Distance along x-axis: x₂ − x₁ = 7 − 3 = 4

Distance along y-axis: y₁ − y₂ = 4 − 1 = 3

Using Baudhāyana–Pythagoras Theorem:

AD = √(4² + 3²) = √(16 + 9) = √25 = 5 units

Side AM is parallel to the x-axis because both points A(0, −2) and M(−5, −2) have the same y-coordinate of −2.

Side MP is parallel to the y-axis because both points M(−5, −2) and P(−5, 2) have the same x-coordinate of −5.

The distance between points (x₁, y₁) and (x₂, y₂) is given by:

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]

This formula is derived from the Baudhāyana–Pythagoras Theorem.

For M to be the midpoint of ST:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

M = ((−3 + 3)/2, (0 + 0)/2) = (0, 0) ✓

The relationship is: If M is the midpoint of ST, then:

x-coordinate of M = (x-coordinate of S + x-coordinate of T)/2

y-coordinate of M = (y-coordinate of S + y-coordinate of T)/2

Since the x-coordinate of point P is −5, point P can lie in Quadrant II or Quadrant III (where the x-coordinate is negative).

General form of coordinates: P(−5, y) where:

• If y > 0: P lies in Quadrant II, coordinates are (−5, positive number)

• If y < 0: P lies in Quadrant III, coordinates are (−5, negative number)

• If y = 0: P lies on the negative x-axis, coordinates are (−5, 0)

If D, E, F are midpoints of sides BC, CA, and AB respectively:

Let A(x₁, y₁), B(x₂, y₂), C(x₃, y₃)

D is midpoint of BC: D = ((x₂+x₃)/2, (y₂+y₃)/2) = (5, 1)

E is midpoint of CA: E = ((x₃+x₁)/2, (y₃+y₁)/2) = (6, 5)

F is midpoint of AB: F = ((x₁+x₂)/2, (y₁+y₂)/2) = (0, 3)

Adding all three equations:

2(x₁+x₂+x₃) = 5+6+0 = 11

2(y₁+y₂+y₃) = 1+5+3 = 9

From E: x₃+x₁ = 12, y₃+y₁ = 10

From F: x₁+x₂ = 0, y₁+y₂ = 6

Solving: x₁ = 7, y₁ = 4

Therefore, A(7, 4)

In a city grid system with N-S and E-W roads:

1. Two perpendicular roads serve as axes (like x and y axes)

2. The city center acts as the origin (0, 0)

3. Streets parallel to these roads are at uniform distances

4. Each street intersection is located using two numbers: position along N-S direction and position along E-W direction

5. A merchant can find a shop by counting units from the city center in both directions

6. For example, intersection (2, 5) represents the point where the 2nd N-S street meets the 5th E-W street

This is exactly how the Sindhu-Sarasvatī Civilization used coordinate systems for urban planning 5000 years ago.

To verify if ABCD is a square, calculate the lengths of all sides:

AB = √[(−1−2)² + (2−1)²] = √[9 + 1] = √10

BC = √[(−2−(−1))² + (−1−2)²] = √[1 + 9] = √10

CD = √[(1−(−2))² + (−2−(−1))²] = √[9 + 1] = √10

DA = √[(2−1)² + (1−(−2))²] = √[1 + 9] = √10

All sides are equal (length √10). Checking if adjacent sides are perpendicular by checking diagonals:

AC = √[(−2−2)² + (−1−1)²] = √[16 + 4] = √20

BD = √[(1−(−1))² + (−2−2)²] = √[4 + 16] = √20

Diagonals are equal, so ABCD is a square.

Area = (side)² = (√10)² = 10 square units

Historical Development of Coordinate Geometry:

1. Sindhu-Sarasvatī Civilization (thousands of years ago): Used grid-based city planning with N-S and E-W streets at uniform distances of ~10 meters.

2. Baudhāyana (c. 800 CE): Developed geometric constructions using E-W and N-S lines, laying the foundation of coordinate geometry through the Baudhāyana–Pythagoras Theorem.

3. Āryabhaṭa (c. 499 CE): Replaced Greek 'chords' with 'sines' for easier coordinate calculations. Mapped the sky using celestial coordinates measured from the ecliptic.

4. Brahmagupta (c. 628 CE): Formalized zero and negative numbers as algebraic entities, enabling the concept of origin (zero) and negative axes in the four-quadrant Cartesian plane.

5. Al-Bīrūnī (c. 1000 CE): Traveled to India, studied the Siddhāntas, and used Indian trigonometric methods to calculate city coordinates across Asia.

6. Ömar Khayyām (c. 1100 CE): First mathematician to solve algebraic problems using geometry by interpreting them as coordinates in the plane.

7. René Descartes (1637 CE): Formalized that any point in 2-D can be defined by two numbers representing distances from perpendicular axes.

Width of the door:

Both points have y-coordinate 0, so the door is on the x-axis.

Width = |x₂ − x₁| = |11.5 − 8.5| = 3 feet

Comfort and Accessibility Analysis:

1. The door width of 3 feet (approximately 0.91 meters) is a standard residential door width.

2. Standard room doors are typically 2.5-3 feet wide.

3. For wheelchair accessibility: Standard wheelchair width is about 2.5-3 feet, so a person in a wheelchair can pass through but with minimal clearance.

4. Ideally, for better accessibility and ease of movement with furniture, a wider door (3.5-4 feet) would be more comfortable.

5. This width allows single entry but makes maneuvering difficult for people with mobility aids.

Conclusion: While the door width meets minimum standards, a slightly wider door would improve accessibility and comfort.