line r has an equation of y + 10 = -2(x - 6). line s is perpendicular to line r and passes through (1, 2)…

Explanation:

Step1: Find slope of line r

The equation of line ( r ) is in point - slope form ( y - y_1=m(x - x_1) ), where ( m ) is the slope. For ( y + 10=-2(x - 6) ), the slope of line ( r ), ( m_r=-2 ).

Step2: Find slope of line s

If two lines are perpendicular, the product of their slopes is ( - 1 ). Let the slope of line ( s ) be ( m_s ). Then ( m_r\times m_s=-1 ). Substituting ( m_r = - 2 ), we get ( - 2\times m_s=-1 ). Solving for ( m_s ), we divide both sides by ( - 2 ): ( m_s=\frac{-1}{-2}=\frac{1}{2} ).

Step3: Use point - slope form to find equation of line s

The point - slope form of a line is ( y - y_1=m(x - x_1) ), where ( (x_1,y_1)=(1,2) ) and ( m = \frac{1}{2} ). Substituting these values, we have ( y - 2=\frac{1}{2}(x - 1) ).

Step4: Convert to slope - intercept form (( y=mx + b ))

Expand the right - hand side: ( y - 2=\frac{1}{2}x-\frac{1}{2} ). Then add 2 to both sides. ( y=\frac{1}{2}x-\frac{1}{2}+2 ). Since ( 2=\frac{4}{2} ), we have ( y=\frac{1}{2}x-\frac{1}{2}+\frac{4}{2}=\frac{1}{2}x+\frac{3}{2} ).

Answer:

( y=\frac{1}{2}x+\frac{3}{2} )