Triangle ABC, AB=8, BC=10, and AC=12, M is the midpoint of AB, and N is the midpoint of BC. What is the length of MN?

Triangle ABC is similar to triangle MBN. Therefore, the corresponding sides are proportional:

MN/AC = BM/AB,
MN/12 = 4/8,
Cross multiply:
8MN = 48,
MN = 6.

By mid point theorem, mn would equal half ac and would be parallel to ac. Thus mn equals 6. The lengths of ab and bc are not required to answer this question. It can be solved by similarity or basic proportionality theorem also.

Thanks monkeys

To find the length of segment MN, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:

Midpoint(M) = ((x1 + x2) / 2, (y1 + y2) / 2)

In this case, let's consider the coordinates of points A, B, and C to find the coordinates of M and N.

Given: AB = 8, BC = 10, and AC = 12

Since M is the midpoint of AB, we know that AM = BM = AB / 2 = 4.

Since N is the midpoint of BC, we know that BN = NC = BC / 2 = 5.

Now, let's consider the coordinates of points A, B, and C.

Assuming that point A is the origin (0,0), we can assign coordinates to points B and C:

Point B: (AB, 0) = (8, 0)
Point C: (x, y)

Since AC = 12, we can find the coordinates of point C using the distance formula:

AC^2 = (x2 - x1)^2 + (y2 - y1)^2

Simplifying, we have:

12^2 = (x - 0)^2 + (y - 0)^2
144 = x^2 + y^2

But we also know that BC = 10, so using the distance formula for BC, we have:

BC^2 = (x2 - x1)^2 + (y2 - y1)^2
10^2 = (x - 8)^2 + (y - 0)^2
100 = (x - 8)^2 + y^2

From these two equations, we can solve for x and y simultaneously.

Simplifying the first equation, we have:

144 = x^2 + y^2

Rearranging the second equation, we have:

100 = (x - 8)^2 + y^2
100 = x^2 - 16x + 64 + y^2

Substituting the first equation into the second equation, we have:

100 = 144 - 16x + 64

Simplifying, we get:

-108 = -16x

Dividing both sides by -16:

6.75 = x

Now substitute this value of x back into the first equation:

144 = (6.75)^2 + y^2
144 = 45.5625 + y^2
98.4375 = y^2

Taking the square root of both sides:

9.9217 = y

So, the coordinates of point C are approximately (6.75, 9.9217).

Now, let's find the coordinates of points M and N:

M is the midpoint of AB, so its x-coordinate is the average of the x-coordinates of A and B, which is (0 + 6.75) / 2 = 3.375.
Since M lies on the y-axis (0, y), its y-coordinate is 0.

Therefore, the coordinates of point M are approximately (3.375, 0).

Similarly, N is the midpoint of BC, so its x-coordinate is the average of the x-coordinates of B and C, which is (6.75 + 8) / 2 = 7.375.
Since N lies on the y-axis (0, y), its y-coordinate is 0.

Therefore, the coordinates of point N are approximately (7.375, 0).

Now, let's find the length of MN using the distance formula:

MN = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(7.375 - 3.375)^2 + (0 - 0)^2]
= √[4^2 + 0^2]
= √[16 + 0]
= √16
= 4

So, the length of segment MN is 4 units.