Question

The point M(3,1.25) lies along the directed line segment from A(-3,5) to B(5,0). Point M divides the segment into a ratio of ________ : _________.

A) 1:4
B) 2:3
C) 3:2
D) 4:1

Answer 1

The point M(3,1.25) lies along the directed line segment from A(-3,5) to B(5,0). Point M divides the segment into a ratio of 3: 2. The correct answer is C. 3:2

Step-by-step explanation:

To find the ratio in which point M divides the line segment AB, we can use the concept of the slope of a line. The slope of the line passing through points A and B can be found using the formula:

m = (y2 - y1) / (x2 - x1)

In this case, A(-3,5) and B(5,0). Plugging the values into the formula, we get:

m = (0 - 5) / (5 - (-3)) = -5 / 8

Now, we can find the equation of the line using the slope and one of the points, say A.

The equation of the line is:

y - 5 = -5/8(x - (-3))

Next, we substitute the x-coordinate of point M (3) into the equation and solve for y to find the y-coordinate of point M:

y - 5 = -5/8(3 - (-3))

Simplifying further:

y - 5 = -5/8(6)

y - 5 = -15/8

y = -15/8 + 5

y = -15/8 + 40/8 = 25/8 = 3.125

Hence, point M has the coordinates (3, 3.125).

Now, we can find the ratio in which M divides the line segment AB by calculating the distances between points:

AM = sqrt((3 - (-3))^2 + (3.125 - 5)^2) = sqrt(36 + 2.890625) ≈ sqrt(38.890625)

MB = sqrt((5 - 3)^2 + (0 - 3.125)^2) = sqrt(2^2 + 9.765625) ≈ sqrt(13.765625)

So, the ratio of AM to MB is approximately:

AM : MB ≈ sqrt(38.890625) : sqrt(13.765625)

AM : MB ≈ 6.23096 : 3.71284

Rounding to the nearest whole number ratio, we get:

AM : MB ≈ 6 : 4 = 3 : 2

Therefore the correct answer is C. 3:2

Answer 2

Point M(3,1.25) divides the segment from A(-3,5) to B(5,0) in the ratio 3:2, making option C the correct answer.

Step-by-step explanation:

The student enquired about the ratio in which point M divides the line segment from point A to point B. Using the coordinates given for points A(-3,5), B(5,0), and M(3,1.25), we can use the section formula to find the desired ratio. The section formula states that if a point M divides a line segment AB in the ratio of m:n, then the coordinates of point M can be determined using the following equations:

Mx = (n × Ax + m × Bx) / (m + n)

My = (n × Ay + m × By) / (m + n)

Substituting the coordinates into the formula and solving for m and n, we find that the ratio in which point M divides the segment AB is 3:2. This means option C) 3:2 is the correct choice.