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Zorluk: OrtaCoordinate Geometry and Lines

In the xyxy-plane, line kk passes through the points (−6,0)(-6, 0) and (0,4)(0, 4). Line mm is perpendicular to line kk and intersects line kk at its yy-intercept. What is the xx-intercept of line mm?

  1. A
    −83-\frac{8}{3}
  2. B
    −6-6
  3. C
    32\frac{3}{2}
  4. 83\frac{8}{3}Cevap
  5. E
    66

Cevap

The xx-intercept of line mm is 83\frac{8}{3}.
Line kk has a slope of 4−00−(−6)=23\frac{4 - 0}{0 - (-6)} = \frac{2}{3}. Because line mm is perpendicular to line kk, its slope is the negative reciprocal, −32-\frac{3}{2}. Line mm intersects line kk at its yy-intercept (0,4)(0, 4), giving line mm the equation y=−32x+4y = -\frac{3}{2}x + 4. Setting y=0y = 0 yields 0=−32x+40 = -\frac{3}{2}x + 4, which solves to x=83x = \frac{8}{3}.

Adım Adım Çözüm

1
Calculate the slope of line kk.
The slope mk=4−00−(−6)=46=23m_k = \frac{4 - 0}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}.
The slope of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is calculated using y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}.
2
Determine the slope and yy-intercept of line mm.
The slope of line mm is mm=−32m_m = -\frac{3}{2}, and its yy-intercept is (0,4)(0, 4).
Perpendicular lines have slopes that are negative reciprocals of each other, and line mm intersects line kk at (0,4)(0, 4).
3
Write the equation of line mm and solve for its xx-intercept.
Setting y=0y = 0 in y=−32x+4y = -\frac{3}{2}x + 4 yields 0=−32x+4  ⟹  32x=4  ⟹  x=830 = -\frac{3}{2}x + 4 \implies \frac{3}{2}x = 4 \implies x = \frac{8}{3}.
The xx-intercept of a line is the value of xx when y=0y = 0.

Anahtar Kavram

Perpendicular line slopes and intercept calculations
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Konular

Coordinate Geometry and Lines

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