The missing coordinate of point A is x = -7 and of point D is y = 11
Step-by-step explanation:
The product of the slopes of perpendicular lines is -1
Example: If two lines are perpendicular and the slope of one of
them is m, then the slope of the other is [tex]\frac{-1}{m}[/tex]
∵ Line Segment BC has endpoints B (3 , 5) and C (7 , 15)
∴ [tex]x_{1}[/tex] = 3 and [tex]x_{2}[/tex] = 7
∴ [tex]y_{1}[/tex] = 5 and [tex]y_{2}[/tex] = 15
- Substitute them in the formula of the slope to find the slope of BC
∴ [tex]m_{BC}=\frac{15-5}{7-3}=\frac{10}{4}[/tex]
- Divide up and down by 2 to reduce it to its simplest form
∴ [tex]m_{BC}=\frac{5}{2}[/tex]
∵ AB ⊥ BC
- Use the rule of the slopes of the perpendicular lines above,
reciprocal the slope of BC and change its sign
∴ [tex]m_{AB}=\frac{-2}{5}[/tex]
∵ A = (x , 9) and B = (3 , 5)
∴ [tex]x_{1}[/tex] = x and [tex]x_{2}[/tex] = 3
∴ [tex]y_{1}[/tex] = 9 and [tex]y_{2}[/tex] = 5
- Substitute them in the formula of the slope to find the slope of AB
∴ [tex]m_{AB}=\frac{5-9}{3-x}=\frac{-4}{3-x}[/tex]
∵ [tex]m_{AB}=\frac{-2}{5}[/tex]
- Equate the two values of the slope of line AB
∴ ∴ [tex]\frac{-4}{3-x}=\frac{-2}{5}[/tex]
- By using cross multiplication
∴ -4 × 5 = -2 × (3 - x)
∴ -20 = -6 + 2x
- Add 6 to both sides
∴ -14 = 2x
- Divide both sides by 2
∴ -7 = x
∴ The x-coordinate of point A is -7
∵ CD ⊥ BC
- Use the rule of the slopes of the perpendicular lines above,
reciprocal the slope of BC and change its sign
∴ [tex]m_{CD}=\frac{-2}{5}[/tex]
∵ C = (7 , 15) and D = (17 , y)
∴ [tex]x_{1}[/tex] = 7 and [tex]x_{2}[/tex] = 17
∴ [tex]y_{1}[/tex] = 15 and [tex]y_{2}[/tex] = y
- Substitute them in the formula of the slope to find the slope of CD
∴ [tex]m_{CD}=\frac{y-15}{17-7}=\frac{y-15}{10}[/tex]
∵ [tex]m_{CD}=\frac{-2}{5}[/tex]
- Equate the two values of the slope of line AB
∴ ∴ [tex]\frac{y-15}{10}=\frac{-2}{5}[/tex]
- By using cross multiplication
∴ 5 × (y - 15) = -2 × 10
∴ 5y - 75 = -20
- Add 75 to both sides
∴ 5y = 55
- Divide both sides by 5
∴ y = 11
∴ The y-coordinate of point D is 11
The missing coordinate of point A is x = -7 and of point D is y = 11
Learn more:
You can learn more about the slopes of the perpendicular lines in brainly.com/question/2601054
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