Find the best solutions to your problems with the help of IDNLearn.com's experts. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.
Sagot :
To determine the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex], given that [tex]\( \overleftrightarrow{AB} \)[/tex] and [tex]\( \overleftrightarrow{BC} \)[/tex] form a right angle at point [tex]\( B \)[/tex], and the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A = (-3, -1) \)[/tex] and [tex]\( B = (4, 4) \)[/tex], follow these steps:
1. Calculate the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]:
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting [tex]\(A\)[/tex] and [tex]\(B\)[/tex] coordinates:
[tex]\[ m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7} \][/tex]
2. Determine the slope of [tex]\( \overleftrightarrow{BC} \)[/tex]:
Because [tex]\( \overleftrightarrow{AB} \)[/tex] and [tex]\( \overleftrightarrow{BC} \)[/tex] are perpendicular, the slope [tex]\( m \)[/tex] of [tex]\( \overleftrightarrow{BC} \)[/tex] is the negative reciprocal of the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]:
[tex]\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5} \][/tex]
3. Find the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex]:
The equation of a line in point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\( B = (4, 4) \)[/tex] and [tex]\( m_{BC} = -\frac{7}{5} \)[/tex]:
[tex]\[ y - 4 = -\frac{7}{5}(x - 4) \][/tex]
4. Convert to slope-intercept form:
Solving for [tex]\( y \)[/tex]:
[tex]\[ y - 4 = -\frac{7}{5} x + \frac{28}{5} \][/tex]
[tex]\[ y = -\frac{7}{5}x + \frac{28}{5} + 4 \][/tex]
[tex]\[ y = -\frac{7}{5}x + \frac{28}{5} + \frac{20}{5} \][/tex]
[tex]\[ y = -\frac{7}{5}x + \frac{48}{5} \][/tex]
5. Convert to general form [tex]\( ax + by = c \)[/tex]:
Multiply through by 5 to clear the fraction:
[tex]\[ 5y = -7x + 48 \][/tex]
Rearrange to general form:
[tex]\[ 7x + 5y = 48 \][/tex]
Compare to the given options:
- Option A: [tex]\( x + 3y = 16 \)[/tex]
- Option B: [tex]\( 2x + y = 12 \)[/tex]
- Option C: [tex]\( -7x - 5y = -48 \)[/tex]
- Option D: [tex]\( 7x - 5y = 48 \)[/tex]
None of the options match [tex]\( 7x + 5y = 48 \)[/tex].
Therefore, the equation of [tex]\( \overleftrightarrow{BC} \)[/tex], given the points and right angle at [tex]\( B \)[/tex], is confirmed by running through the correct procedures and calculations as:
[tex]\[ \boxed{\text{None}} \][/tex]
1. Calculate the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]:
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting [tex]\(A\)[/tex] and [tex]\(B\)[/tex] coordinates:
[tex]\[ m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7} \][/tex]
2. Determine the slope of [tex]\( \overleftrightarrow{BC} \)[/tex]:
Because [tex]\( \overleftrightarrow{AB} \)[/tex] and [tex]\( \overleftrightarrow{BC} \)[/tex] are perpendicular, the slope [tex]\( m \)[/tex] of [tex]\( \overleftrightarrow{BC} \)[/tex] is the negative reciprocal of the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]:
[tex]\[ m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5} \][/tex]
3. Find the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex]:
The equation of a line in point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\( B = (4, 4) \)[/tex] and [tex]\( m_{BC} = -\frac{7}{5} \)[/tex]:
[tex]\[ y - 4 = -\frac{7}{5}(x - 4) \][/tex]
4. Convert to slope-intercept form:
Solving for [tex]\( y \)[/tex]:
[tex]\[ y - 4 = -\frac{7}{5} x + \frac{28}{5} \][/tex]
[tex]\[ y = -\frac{7}{5}x + \frac{28}{5} + 4 \][/tex]
[tex]\[ y = -\frac{7}{5}x + \frac{28}{5} + \frac{20}{5} \][/tex]
[tex]\[ y = -\frac{7}{5}x + \frac{48}{5} \][/tex]
5. Convert to general form [tex]\( ax + by = c \)[/tex]:
Multiply through by 5 to clear the fraction:
[tex]\[ 5y = -7x + 48 \][/tex]
Rearrange to general form:
[tex]\[ 7x + 5y = 48 \][/tex]
Compare to the given options:
- Option A: [tex]\( x + 3y = 16 \)[/tex]
- Option B: [tex]\( 2x + y = 12 \)[/tex]
- Option C: [tex]\( -7x - 5y = -48 \)[/tex]
- Option D: [tex]\( 7x - 5y = 48 \)[/tex]
None of the options match [tex]\( 7x + 5y = 48 \)[/tex].
Therefore, the equation of [tex]\( \overleftrightarrow{BC} \)[/tex], given the points and right angle at [tex]\( B \)[/tex], is confirmed by running through the correct procedures and calculations as:
[tex]\[ \boxed{\text{None}} \][/tex]
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.
