Title 59 Lines and Distances.pdf ✅

MSTC 59: Lines and Distances 1. Line Properties 1.1. Slope Formula [DERIVATION] tan θ = y2 − y1 x2 − x1 Therefore, the slope is m = tan θ = rise run = y2 − y1 x2 − x1 • A vertical line has an undefined slope. • A horizontal line has a 0 slope. • The higher the slope value, the steeper the line. • If the line rises to the right, it has a positive slope. • If the line lowers to the right, it has a negative slope. 1.2. Division of Line Segment This segment focuses on the determination of the coordinates of the division points of a line segment. 1.2.1. Midpoint The midpoint of a line is a point exactly midway through a line segment. The formula is (xm, ym) = ( x2 + x1 2 , y1 + y2 2 )
1.2.2. Internal and External Division points Ratio and proportion may be used instead of derived formulas to simplify the solving process and eliminate possible errors due to inaccurate substitution of values. Here, any of the points in the figure can be division points. Using ratio and proportion, x2 − x3 x3 − x1 = m n = y2 − y3 y3 − y1 Any unknown coordinates can be determined using a portion of the equation. 1.3. Parallel Lines Parallel lines in Euclidean Space never intersect. This condition is only possible if and only if m1 = m2 Here, m1 and m2 are slopes of two parallel lines. 1.4. Perpendicular Lines The product of slopes of two perpendicular lines is always −1. [PROOF] From the slope definition, tan θ = m1 tan(θ + 90°) = m2 From trigonometric identities, tan(θ + 90°) = − cot θ m2 = − 1 m1 m2m1 = −1 Hence, proven.

2.2. Point-slope Form This form is helpful when the slope of the line and a point within the line are known. [DERIVATION] From the two-point form, y − y1 = ( y2 − y1 x2 − x1 ) (x − x1 ) But since the slope is known, and it is defined that m = y2 − y1 x2 − x1 Then, the point-slope form is y − y1 = m(x − x1 ) 2.3. Slope-intercept Form This form is one of the most commonly used forms. This form directly shows the slope and the y-intercept of the line. Given that m is the slope and b is the y-intercept, then y = mx + b [LINK TO POINT-SLOPE FORM] From the point-slope form, y − y1 = m(x − x1 ) y = mx − mx1 + y1 Setting b = −mx1 + y1 gives y = mx + b 2.4. Intercept Form The intercept form features the two intercepts of the line. If the x-intercept is at (a, 0) and the y- intercept at (0, b), then from the point-slope form, [DERIVATION] y − y1 = ( y2 − y1 x2 − x1 ) (x − x1 ) y − b = b − 0 0 − a (x − 0) y − b = − b a x bx + ay = ab Dividing both sides by ab results in the intercept form x a + y b = 1