The midpoint of the line segment joining (2a,4) and (-2,2b) is (1,2a+1). Find the values of a and b .

To find the values of a and b, where the midpoint of the line segment joining points (2a, 4) and (-2, 2b) is point (1, 2a+1), we can use the midpoint formula:

[tex]\boxed{\begin{array}{c}\underline{\sf Midpoint \;formula}\\\\M(x,y) =\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)\\\\\textsf{where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.}\\\end{array}}[/tex]

In this case:

M = (1, 2a+1)
(x₁, y₁) = (2a, 4)
(x₂, y₂) = (-2, 2b)

Substitute the coordinates into the midpoint formula:

[tex](1, 2a+1) =\left(\dfrac{-2+2a}{2},\dfrac{2b+4}{2}\right)[/tex]

We can simplify the right side of the equation since the numerators are divisible by 2:

[tex](1, 2a+1) =\left(-1+a,b+2\right)[/tex]

Equate the x-coordinates and solve for a:

[tex]1=-1+a\\\\a=1-(-1)\\\\a=2[/tex]

Equate the y-coordinates, substitute a = 2, and solve for b:

[tex]2a+1=b+2\\\\2(2)+1=b+2\\\\4+1=b+2\\\\5=b+2\\\\b=5-2\\\\b=3[/tex]

Therefore, the values of a and b are:

[tex]\Large\boxed{\boxed{\begin{array}{l}a=2\\b=3\end{array}}}[/tex]