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Zorluk: OrtaCoordinate Geometry and Lines

In the xyxy-plane, line kk passes through the points (−2,5)(-2, 5) and (4,−7)(4, -7). Which of the following statements about line kk must be true? Select all that apply.

  1. Line kk has a slope of −2-2.Cevap
  2. Line kk passes through the point (3,−5)(3, -5).Cevap
  3. Line kk does not pass through Quadrant III.Cevap
  4. D
    Line kk has an xx-intercept at (−2,0)(-2, 0).
  5. E
    A line perpendicular to line kk has a slope of 22.

Cevap

The correct statements are: line kk has a slope of −2-2, line kk passes through the point (3,−5)(3, -5), and line kk does not pass through Quadrant III.
The slope of line kk is m=−7−54−(−2)=−2m = \frac{-7 - 5}{4 - (-2)} = -2. The line equation is y=−2x+1y = -2x + 1. Substituting x=3x = 3 yields y=−5y = -5, confirming (3,−5)(3, -5) is on the line. For all negative xx-values (x<0x < 0), y=−2x+1y = -2x + 1 remains strictly positive (y>1y > 1), so no point on the line has both negative coordinates; hence line kk never passes through Quadrant III.

Adım Adım Çözüm

1
Calculate the slope of line kk
slope m=−7−54−(−2)=−126=−2m = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2
The slope formula is m=ΔyΔx=y2−y1x2−x1m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}.
2
Determine the equation of line kk
y=−2x+1y = -2x + 1
Using point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (−2,5)(-2, 5) gives y−5=−2(x+2)  ⟹  y=−2x+1y - 5 = -2(x + 2) \implies y = -2x + 1.
3
Test point (3,−5)(3, -5) and calculate the xx-intercept
Point (3,−5)(3, -5) lies on the line since −2(3)+1=−5-2(3) + 1 = -5. Setting y=0y = 0 gives 0=−2x+1  ⟹  x=120 = -2x + 1 \implies x = \frac{1}{2}.
A point lies on a line if its coordinates satisfy the line equation. The xx-intercept occurs where y=0y = 0.
4
Analyze quadrant passage and perpendicular slope
For x<0x < 0, y=−2x+1>0y = -2x + 1 > 0, so points in Quadrant III (x<0,y<0x < 0, y < 0) are never reached. Perpendicular slope is −1−2=12-\frac{1}{-2} = \frac{1}{2}.
Quadrant III requires both coordinates to be negative. Perpendicular lines have negative reciprocal slopes.

Anahtar Kavram

Coordinate Geometry of Lines: Slope, Line Equations, Intercepts, Quadrant Passage, and Perpendicular Slopes
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Konular

Coordinate Geometry and Lines

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