Coordinate Geometry — Andhra Pradesh (SSC) Class 9 Mathematics Solutions (Free)

Step 1: Understand quadrants Quadrant I: x > 0, y > 0 Quadrant II: x < 0, y > 0 Quadrant III: x < 0, y < 0 Quadrant IV: x > 0, y < 0 Step 2: Classify each point Point A(3, 4): x = 3 > 0, y = 4 > 0 → Quadrant I Point B(-2, 5): x = -2 < 0, y = 5 > 0 → Quadrant II Point C(-3, -2): x = -3 < 0, y = -2 < 0 → Quadrant III Point D(4, -1): x = 4 > 0, y = -1 < 0 → Quadrant IV Step 3: Plot on coordinate plane - Mark origin O at (0, 0) - A is 3 units right, 4 units up - B is 2 units left, 5 units up - C is 3 units left, 2 units down - D is 4 units right, 1 unit down Answer: A: Quadrant I, B: Quadrant II, C: Quadrant III, D: Quadrant IV

Points: P(1, 3) and Q(4, 7) Use distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] Step 1: Identify coordinates x₁ = 1, y₁ = 3 x₂ = 4, y₂ = 7 Step 2: Calculate differences x₂ - x₁ = 4 - 1 = 3 y₂ - y₁ = 7 - 3 = 4 Step 3: Square the differences (x₂ - x₁)² = 3² = 9 (y₂ - y₁)² = 4² = 16 Step 4: Sum and take square root d = √(9 + 16) = √25 = 5 Answer: Distance PQ = 5 units

Points: A(-2, 4) and B(6, -2) Use midpoint formula: M = [(x₁ + x₂)/2, (y₁ + y₂)/2] Step 1: Identify coordinates x₁ = -2, y₁ = 4 x₂ = 6, y₂ = -2 Step 2: Calculate x-coordinate of midpoint x_M = (x₁ + x₂)/2 = (-2 + 6)/2 = 4/2 = 2 Step 3: Calculate y-coordinate of midpoint y_M = (y₁ + y₂)/2 = (4 + (-2))/2 = 2/2 = 1 Step 4: Write midpoint coordinates M = (2, 1) Answer: The midpoint is (2, 1)

Points: A(0, 0), B(3, 4), C(6, 8) Points are collinear if they lie on the same line. Method 1: Using slope Slope between A and B: m_AB = (4 - 0)/(3 - 0) = 4/3 Slope between B and C: m_BC = (8 - 4)/(6 - 3) = 4/3 Slope between A and C: m_AC = (8 - 0)/(6 - 0) = 8/6 = 4/3 Since all three slopes are equal (4/3), the points are collinear. Method 2: Using area of triangle If area of triangle ABC = 0, then points are collinear. Area = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| = (1/2)|0(4 - 8) + 3(8 - 0) + 6(0 - 4)| = (1/2)|0 + 24 - 24| = (1/2)|0| = 0 Since area = 0, the points are collinear. Conclusion: A, B, and C are collinear.

Vertices: A(1, 2), B(4, 2), C(4, 5) Method 1: Using coordinate formula Area = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| Step 1: Identify coordinates x₁ = 1, y₁ = 2 x₂ = 4, y₂ = 2 x₃ = 4, y₃ = 5 Step 2: Substitute in formula Area = (1/2)|1(2 - 5) + 4(5 - 2) + 4(2 - 2)| = (1/2)|1(-3) + 4(3) + 4(0)| = (1/2)|-3 + 12 + 0| = (1/2)|9| = 9/2 = 4.5 Method 2: Geometric observation Base AB = distance from A(1,2) to B(4,2) = 4 - 1 = 3 Height = perpendicular distance from C(4,5) to line AB = 5 - 2 = 3 Area = (1/2) × base × height = (1/2) × 3 × 3 = 4.5 Answer: Area of triangle = 4.5 square units

Given: Point lies on x-axis, distance from (2, 3) is 5 Step 1: Represent the point Since point is on x-axis, coordinates are (x, 0) for some value of x. Step 2: Apply distance formula Distance = √[(x - 2)² + (0 - 3)²] = 5 Step 3: Square both sides (x - 2)² + 9 = 25 Step 4: Simplify (x - 2)² = 25 - 9 (x - 2)² = 16 Step 5: Take square root x - 2 = ±4 x = 2 + 4 or x = 2 - 4 x = 6 or x = -2 Step 6: Find both points Point 1: (6, 0) Point 2: (-2, 0) Step 7: Verify Distance from (6, 0) to (2, 3) = √[(6-2)² + (0-3)²] = √[16 + 9] = √25 = 5 ✓ Distance from (-2, 0) to (2, 3) = √[(-2-2)² + (0-3)²] = √[16 + 9] = √25 = 5 ✓ Answer: The points are (6, 0) and (-2, 0)

Points: P(-1, 2) and Q(3, -4) Step 1: Apply slope formula Slope m = (y₂ - y₁)/(x₂ - x₁) = (-4 - 2)/(3 - (-1)) Step 2: Calculate numerator = (-6)/(3 + 1) = -6/4 Step 3: Simplify = -3/2 Step 4: Interpret the slope Slope m = -3/2 < 0 (negative) When slope is negative, the line is falling (decreasing from left to right). Interpretation: For every 2 units moved to the right, the line drops 3 units down. Answer: Slope = -3/2; The line is falling.

Vertices: A(0, 0), B(4, 0), C(4, 3), D(0, 3) Step 1: Calculate all four sides using distance formula AB = √[(4-0)² + (0-0)²] = √16 = 4 BC = √[(4-4)² + (3-0)²] = √9 = 3 CD = √[(0-4)² + (3-3)²] = √16 = 4 DA = √[(0-0)² + (0-3)²] = √9 = 3 Step 2: Check opposite sides are equal AB = CD = 4 ✓ BC = DA = 3 ✓ Opposite sides are equal. Step 3: Check all angles are right angles using slopes Slope of AB = (0-0)/(4-0) = 0 (horizontal) Slope of BC = (3-0)/(4-4) = undefined (vertical) Slope of CD = (3-3)/(0-4) = 0 (horizontal) Slope of DA = (0-3)/(0-0) = undefined (vertical) Step 4: Verify perpendicularity AB is perpendicular to BC (horizontal ⊥ vertical) ✓ BC is perpendicular to CD (vertical ⊥ horizontal) ✓ All angles are 90°. Step 5: Calculate diagonals AC = √[(4-0)² + (3-0)²] = √[16 + 9] = √25 = 5 BD = √[(0-4)² + (3-0)²] = √[16 + 9] = √25 = 5 Diagonals are equal: AC = BD ✓ Conclusion: ABCD is a rectangle (opposite sides equal, all angles 90°, diagonals equal).