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Analytical geometry, help with this

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3 parts
2.1
(6, 3)
2.2

y=3x−15y = 3x - 15

2.3
9

Step-by-step solution

9 steps

Here are the solutions to your analytical geometry problems.

Part 2.1· 2 steps

Determine the coordinates of E.

E is the midpoint of line segment AB. The coordinates of A are (5,0)(5, 0) and B are (7,6)(7, 6).
The midpoint formula is:

E=(x1+x22,y1+y22)E = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

  1. Substitute the coordinates of A and B into the midpoint formula.

    E=(5+72,0+62)E = \left(\frac{5+7}{2}, \frac{0+6}{2}\right)

  2. Calculate the coordinates.

    E=(122,62)E = \left(\frac{12}{2}, \frac{6}{2}\right) E=(6,3)E = (6, 3)

    The coordinates of E are (6,3)\boxed{(6, 3)}.

Part 2.2

y=3x−15y = 3x - 15

Determine the equation of line BA.

To find the equation of a line, we need its slope and a point on the line.
The coordinates of A are $(5, 0)$ and B are $(7, 6)$.

  1. Calculate the slope (mm) of line BA using the formula m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}.

    mBA=6−07−5m_{BA} = \frac{6-0}{7-5} mBA=62m_{BA} = \frac{6}{2} mBA=3m_{BA} = 3

  2. Use the point-slope form of a linear equation, y−y1=m(x−x1)y - y_1 = m(x - x_1), with point A(5,0)(5, 0) and slope mBA=3m_{BA}=3.

    y−0=3(x−5)y - 0 = 3(x - 5)

  3. Simplify the equation.

    y=3x−15y = 3x - 15

    The equation of line BA is y=3x−15\boxed{y = 3x - 15}.

Part 2.39

Line BA is parallel to straight line with equation: rx−3y+5=0rx - 3y + 5 = 0. Calculate the value of r.

If two lines are parallel, their slopes are equal.

  1. The slope of line BA is mBA=3m_{BA} = 3 (from 2.2).

  2. Find the slope of the given line rx−3y+5=0rx - 3y + 5 = 0 by rearranging it into the slope-intercept form y=mx+cy = mx + c.

    rx−3y+5=0rx - 3y + 5 = 0 −3y=−rx−5-3y = -rx - 5 3y=rx+53y = rx + 5 y=r3x+53y = \frac{r}{3}x + \frac{5}{3}

    The slope of this line is m2=r3m_2 = \frac{r}{3}.

  3. Equate the slopes since the lines are parallel.

    mBA=m2m_{BA} = m_2 3=r33 = \frac{r}{3}

  4. Solve for rr.

    r=3×3r = 3 \times 3 r=9r = 9

    The value of r is 9\boxed{9}.

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