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Find the equation of the perpendicular bisector of the line segment joining given points:
(a) (2, 4) and (8, 10)
(b) (3, -7) and (-5, 3)

ANSWERS:
a. x+y =12
b. 4x-5y-6=0​

Respuesta :

Sure, I can help you with that! Let's find the equation of the perpendicular bisector for each line segment.

(a) The midpoint of the line segment joining (2, 4) and (8, 10) is ((2+8)/2, (4+10)/2) which simplifies to (5, 7). The slope of the line segment is (10-4)/(8-2) = 6/6 = 1.

Since the perpendicular bisector has a negative reciprocal slope, the slope of the perpendicular bisector is -1. Using the point-slope form of a line, we have:

y - 7 = -1(x - 5)

Simplifying the equation, we get:

y - 7 = -x + 5

y = -x + 12

So, the equation of the perpendicular bisector is y = -x + 12.

(b) The midpoint of the line segment joining (3, -7) and (-5, 3) is ((3+(-5))/2, (-7+3)/2) which simplifies to (-1, -2). The slope of the line segment is (3-(-7))/(-5-3) = 10/-8 = -5/4.

Since the perpendicular bisector has a negative reciprocal slope, the slope of the perpendicular bisector is 4/5. Using the point-slope form of a line, we have:

y - (-2) = 4/5(x - (-1))

Simplifying the equation, we get:

y + 2 = 4/5(x + 1)

Multiplying both sides by 5 to eliminate the fraction, we get:

5y + 10 = 4x + 4

Rearranging the equation, we have:

4x - 5y - 6 = 0

So, the equation of the perpendicular bisector is 4x - 5y - 6 = 0.
msm555

Answer:

a.  [tex]\sf x + y = 12 [/tex]

b. [tex]\sf  4x - 5y - 6 = 0 [/tex]

Step-by-step explanation:

To find the equation of the perpendicular bisector of the line segment joining two given points, follow these steps:

(a) Points (2, 4) and (8, 10):

1. Find the midpoint of the line segment:

[tex]\sf  M \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) [/tex]

[tex]\sf  M \left( \dfrac{2 + 8}{2}, \dfrac{4 + 10}{2} \right) [/tex]

[tex]\sf  M (5, 7) [/tex]

2. Find the slope ([tex]\sf  m_{\textsf{original}} [/tex]) of the line through the given points:

[tex]\sf  m_{\textsf{original}} = \dfrac{y_2 - y_1}{x_2 - x_1} [/tex]

[tex]\sf  m_{\textsf{original}} = \dfrac{10 - 4}{8 - 2} [/tex]

[tex]\sf  m_{\textsf{original}} = \dfrac{6}{6} [/tex]

[tex]\sf  m_{\textsf{original}} = 1 [/tex]

3. Find the negative reciprocal of [tex]\sf  m_{\textsf{original}} [/tex] to get the slope of the perpendicular bisector ([tex]\sf  m_{\textsf{bisector}} [/tex]):

[tex]\sf  m_{\textsf{bisector}} = -\dfrac{1}{1} [/tex]

[tex]\sf  m_{\textsf{bisector}} = -1 [/tex]

4. Use the point-slope form to find the equation of the perpendicular bisector:

[tex]\sf  y - y_M = m_{\textsf{bisector}}(x - x_M) [/tex]

[tex]\sf  y - 7 = -1(x - 5) [/tex]

[tex]\sf  y - 7 = -x + 5 [/tex]

[tex]\sf  x + y = 12 [/tex]

So, the equation of the perpendicular bisector is [tex]\sf x + y = 12 [/tex].

(b) Points (3, -7) and (-5, 3):

1. Find the midpoint of the line segment:

[tex]\sf  M \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right) [/tex]

[tex]\sf  M \left( \dfrac{3 + (-5)}{2}, \dfrac{-7 + 3}{2} \right) [/tex]

[tex]\sf  M (-1, -2) [/tex]

2. Find the slope ([tex]\sf  m_{\textsf{original}} [/tex]) of the line through the given points:

[tex]\sf  m_{\textsf{original}} = \dfrac{3 - (-7)}{(-5) - 3} [/tex]

[tex]\sf  m_{\textsf{original}} = \dfrac{10}{-8} [/tex]

[tex]\sf  m_{\textsf{original}} = -\dfrac{5}{4} [/tex]

3. Find the negative reciprocal of [tex]\sf  m_{\textsf{original}} [/tex] to get the slope of the perpendicular bisector ([tex]\sf  m_{\textsf{bisector}} [/tex]):

[tex]\sf  m_{\textsf{bisector}} = \dfrac{4}{5} [/tex]

4. Use the point-slope form to find the equation of the perpendicular bisector:

[tex]\sf  y - y_M = m_{\textsf{bisector}}(x - x_M) [/tex]

[tex]\sf  y - (-2) = \dfrac{4}{5}(x - (-1)) [/tex]

[tex]\sf  y + 2 = \dfrac{4}{5}(x + 1) [/tex]

[tex]\sf  5y + 10 = 4(x + 1) [/tex]

[tex]\sf  5y + 10 = 4x + 4 [/tex]

[tex]\sf  4x - 5y - 6 = 0 [/tex]

So, the equation of the perpendicular bisector is [tex]\sf  4x - 5y - 6 = 0 [/tex].

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