Analytic Geometry: Coordinate Systems and Equations

<p>He established a geometry using a coordinate system. (D)</p> Signup and view all the answers

<p>Abscissa (D)</p> Signup and view all the answers

<p>$-\frac{1}{2}$ (A)</p> Signup and view all the answers

<p>0 (A)</p> Signup and view all the answers

<p>Perpendicular (A)</p> Signup and view all the answers

<p>$3/11$ (C)</p> Signup and view all the answers

<p>Their slopes are equal. (D)</p> Signup and view all the answers

<p>Infinity (B)</p> Signup and view all the answers

<p>A, B, C (C)</p> Signup and view all the answers

<p>Counter-clockwise order only. (A)</p> Signup and view all the answers

<p>Positive, because B is positive and the point is above the line. (A)</p> Signup and view all the answers

<p>$d = \frac{|6 - 12|}{\sqrt{2^2 + 3^2}}$ (D)</p> Signup and view all the answers

<p>$x = \frac{(1 \cdot 2) + (4 \cdot 1)}{2 + 1}$ (B)</p> Signup and view all the answers

<p>(6, 2) (D)</p> Signup and view all the answers

<p>-5/3 (B)</p> Signup and view all the answers

<p>3 (C)</p> Signup and view all the answers

<p>Negative, because B is positive and the point is likely below the line. (C)</p> Signup and view all the answers

<p>(1, -1) (C)</p> Signup and view all the answers

A branch of Mathematics combining Algebra and Geometry through coordinates.

A coordinate system where each point is identified by an ordered pair (x, y).

Formula to find the distance between two points: d = √((x2 - x1)² + (y2 - y1)²).

Extends the distance formula to three dimensions: d = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²).

Calculates how far a point is from a line using the line's equation Ax + By + C = 0.

d = ±√(A² + B²) for line Ax + By + C = 0

Use + if B is positive and point is above/right the line

Use + if B is negative and point is below/left the line

d = |C1 - C2| / ±√(A² + B²) uses absolute difference

Point P divides segment between (x1,y1) and (x2,y2) with ratios r1, r2

x = (x1r2 + x2r1) / (r1 + r2) for divided segment

y = (y1r2 + y2r1) / (r1 + r2) for divided segment

Midpoint coordinates: x = (x1 + x2)/2, y = (y1 + y2)/2

The slope is the rise (∆y) over run (∆x) of a line, m = ∆y/∆x.

A line parallel to the x-axis has a slope of zero (m = 0).

A line parallel to the y-axis has a slope of infinity (m = ∞).

Parallel lines have the same slope (m1 = m2).

For perpendicular lines, the slopes are negative reciprocals (m2 = -1/m1).

The angle θ between lines with slopes m1 and m2 is given by θ = tan⁻¹((m2 - m1)/(1 + m2*m1)).

The area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is determined using counterclockwise order.

Use the determinant method for the area: Area = ½ | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |.