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.

solution:

as, mean is defined as sum of all the observations divided by total number of observations.

and , median is mid value of the data whether arranged in ascending order or descending order.

consider, roadsters:

2,3,4,5,6,7,8,9

mean =

also, median= , because number of observations is even, so median is mean of two mid values.

now, coming to bandits

2,3,4,5,6

mean =  

median = 4, as number of observations is odd.

so, roadsters has greater median as well mean than bandits.

→option a is true about the repairs performed on 2 types of cars:

bandits have lower median and mean than roadsters.

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."