Q is the midpoint of GH¯¯¯¯¯¯ , GQ=2x+3, and GH=5x−5 .

What is the length of GQ¯¯¯¯¯ ?

Question

Q is the midpoint of GH¯¯¯¯¯¯ , GQ=2x+3, and GH=5x−5 .

What is the length of GQ¯¯¯¯¯ ?

Maverick Smith · Accepted Answer

#GQ=25# As #Q# is the midpoint of #GH#, we have #GQ=QH# and #GH=GQ+QH=2xxGQ# Now as #GQ=2x+3#, and #GH=5x−5#, we have #5x-5=2xx(2x+3)# or #5x-5=4x+6# or #5x-4x=6+5# i.e. #x=11# Hence, #GQ=2xx11+3=22+3=25#

Parker Allen · Answer

undefined To find the length of $ GQ $, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints $ (x_1, y_1) $ and $ (x_2, y_2) $ are given by:

$$ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Given that $ Q $ is the midpoint of $ GH $, we can set up the equation:

$$ \left( \frac{x_G + x_H}{2}, \frac{y_G + y_H}{2} \right) = (2x + 3, 5x - 5) $$

Since $ Q $ is the midpoint, its coordinates are the averages of the coordinates of $ G $ and $ H $:

$$ \frac{x_G + x_H}{2} = 2x + 3 $$
$$ \frac{y_G + y_H}{2} = 5x - 5 $$

Given that $ GQ = 2x + 3 $, we can solve the equation for $ x $:

$$ 2x + 3 = GQ = \frac{x_G + x_H}{2} $$
$$ 2x + 3 = \frac{x_G + x_H}{2} $$
$$ 4x + 6 = x_G + x_H $$

Similarly, we can find $ GH $:

$$ 5x - 5 = GH = \frac{y_G + y_H}{2} $$
$$ 10x - 10 = y_G + y_H $$

We can now solve the system of equations to find $ x $:

$$ 4x + 6 = x_G + x_H $$
$$ 10x - 10 = y_G + y_H $$

Now, we can substitute the value of $ x $ into $ GQ $ to find its length:

$$ GQ = 2x + 3 $$