Geometry And Algebra — Important Questions

The midpoint of (a,b) and (c,d) is ((a+c)/2, (b+d)/2), Parallel lines have equal slopes, Distance formula is √[(x₂-x₁)² + (y₂-y₁)²], Perpendicular lines have slopes that multiply to give -1

All statements are correct except 'The slope of a vertical line is zero' - vertical lines have undefined slope, while horizontal lines have zero slope. The other statements are fundamental properties of coordinate geometry.

8

Using point-slope form: slope = (y₂ - y₁)/(x₂ - x₁). Given slope = 3/4, point (2, 5), and x₂ = 6. So 3/4 = (y - 5)/(6 - 2) = (y - 5)/4. Cross-multiplying: 3 × 4 = 4(y - 5), so 12 = 4y - 20, thus 4y = 32, and y = 8.

y = 2x

First find slope: m = (6-2)/(3-1) = 4/2 = 2. Using point-slope form with point (1,2): y - 2 = 2(x - 1), so y - 2 = 2x - 2, therefore y = 2x. We can verify: when x = 1, y = 2; when x = 3, y = 6. Both points satisfy the equation.

(x - 3)² + (y - 4)² = 25

The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. With center (3, 4) and radius 5: (x - 3)² + (y - 4)² = 5² = 25.