📚StudySolv

Class 9 · Mathematics · GANITA MANJARI

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

SolutionsNotesImportant Questions

1 Mark5 questions

Q1.short

What are the coordinates of the origin in a 2-D Cartesian coordinate system?

▶ Show Answer
The coordinates of the origin are (0, 0). The origin is the point of intersection of the x-axis and y-axis.
Q2.short

A point lies on the x-axis. What can you say about its y-coordinate?

▶ Show Answer
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).
Q3.short

In which quadrant does the point P(−3, 5) lie?

▶ Show Answer
The point P(−3, 5) lies in Quadrant II. In Quadrant II, the x-coordinate is negative and the y-coordinate is positive.
Q4.short

Which axis is horizontal and which is vertical in a 2-D Cartesian coordinate system?

▶ Show Answer
The x-axis is horizontal and the y-axis is vertical.
Q5.mcq

Which of the following statements is true about the coordinates (x, y) and (y, x)?

▶ Show Answer
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).

3 Marks6 questions

Q1.short

The coordinates of points A and D are A(3, 4) and D(7, 1). Find the distance AD using the Baudhāyana–Pythagoras Theorem.

▶ Show Answer
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
Q2.short

Consider the points R(3, 0), A(0, −2), M(−5, −2), and P(−5, 2). Identify which side of quadrilateral RAMP is parallel to one of the axes.

▶ Show Answer
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.
Q3.short

Write the formula for finding the distance between two points (x₁, y₁) and (x₂, y₂) in a 2-D plane.

▶ Show Answer
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.
Q4.short

If M(0, 0) is the midpoint of segment ST where S has coordinates (−3, 0) and T has coordinates (3, 0), verify this and state the relationship between the coordinates of M, S, and T.

▶ Show Answer
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
Q5.short

A point P has x-coordinate equal to −5. Predict which quadrants point P can lie in and write the general form of its coordinates.

▶ Show Answer
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)
Q6.short

The midpoints of the sides of triangle ABC are D(5, 1), E(6, 5), and F(0, 3). Using the midpoint relationship, explain how you would find the coordinates of vertex A.

▶ Show Answer
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)

5 Marks4 questions

Q1.long

Explain how the concept of coordinates is used in real-world applications, using the example of a city grid system as described in the chapter.

▶ Show Answer
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.
Q2.long

Plot the points A(2, 1), B(−1, 2), C(−2, −1), and D(1, −2) and determine whether ABCD is a square. Calculate the area if it is a square.

▶ Show Answer
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
Q3.long

Describe the historical development of coordinate geometry, highlighting the contributions of ancient Indian mathematicians mentioned in the chapter.

▶ Show Answer
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.
Q4.long

In Reiaan's room, the door is represented by segment D₁R₁ where D₁ is at (8.5, 0) and R₁ is at (11.5, 0). Find the width of the door. Is this a comfortable width for a room door? Discuss accessibility considerations.

▶ Show Answer
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.