A median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle are drawn below. [asy] pair A, B,C, X,Y, Z; A = (0,0); B = (1,0); C = (0.3,0.8); X = (B+C)/2; Y = (A+C)/2; Z = (A+B)/2; draw(A--X, red); draw(B--Y, red); draw(C--Z, red); draw(A--B--C--A); [/asy] Note that the three medians appear to intersect at the same point! Let's try this out with a particular triangle. Consider the triangle $ABC$ with $A = (3,6)$, $B = (-5,2)$, and $C = (7,-8)$. (a) Let $D,$ $E,$ $F$ be the midpoints of $\overline{BC},$ $\overline{AC},$ $\overline{AB},$ respectively. Find the equations of medians $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ (b) Show that the three medians in part (a) all pass through the same point.

(a) The equation of AD is y = 4.5·x -7.5

The equation of BE is y = -0.3·x + 0.5

The equation of CF is y = -1.5·x + 2.5.

(b) The three equations representing the three lines AD, BE and CF, pass through the same point (which can be written as (1.667, 0)).

Step-by-step explanation:

The given triangle coordinates are;

A = (3, 6)

B = (-5, 2)

C = (7, -8)

The midpoint of BC = D

The midpoint of AC = E

The midpoint of AB = F

Therefore, we have;

The coordinates of point D = (B + C)/2 = (((-5) + 7 )/2, (2 + (-8))/2)

The coordinates of point D =  (1, -3)

The coordinates of point E = (A + C)/2 = ((3 + 7 )/2, (6 + (-8))/2)

The coordinates of point E =  (5, -1)

The coordinates of point F = (A + B)/2 = ((3 + (-5) )/2, (6 + 2)/2)

The coordinates of point F =  (-1, 4)

The equation of AD in slope and intercept form is therefore;

A = (3, 6), D =  (1, -3)

Therefore, we have;

y - 6 = 4.5×(x - 3)

y - 6 = 4.5·x -13.5

y = 4.5·x -13.5 + 6 = 4.5·x -7.5

y = 4.5·x -7.5

The equation of BE in slope and intercept form is found as follows;

B = (-5, 2), E =  (5, -1)

Slope = ((-1)-2)/(5 - (-5)) = -0.3

The point slope equation is y - 2 = -0.3×(x - (-5))

y - 2 = -0.3·x - 1.5

y = -0.3·x - 1.5 + 2 = -0.3·x + 0.5

y = -0.3·x + 0.5

The equation of CF in slope and intercept form is found as follows;

C = (7, -8), F =  (-1, 4)

Slope = (4 - (-8))/((-1) - 7) = -1.5

The point slope equation is y - (-8) = -1.5×(x - 7)

y + 8 = -1.5·x + 10.5

y = -1.5·x + 10.5 - 8

y = -1.5·x + 2.5.

Where CF and BE intersect, we have;

-0.3·x + 0.5 = -1.5·x + 2.5.

1.5·x -0.3·x = 2.5 - 0.5 = 2.0

x = 2.0/1.2 = 5/3

y = -1.5×5/3+ 2.5 = 0

We check for line AD, where y = 4.5·x -7.5

When x = 5/3, we get;

y = 4.5×(5/3) -7.5  = 0

Therefore, the three equations pass through the point (5/3, 0) which is the same point.

g(x) = x² + 2

step-by-step explanation:

it is given that the graphs are the same shape.

the only difference is that the red graph is 2 units higher than the blue graph.

that is to say that the red graph is simply the blue graph that has been moved vertically by 2 units in the positive y direction.

hence

g(x) = f(x) + 2

or

g(x) = x² + 2

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step-by-step explanation: