I need help with my math.. Can someone please help..
In the coordinate plane point X lies on line segment MN. The coordinates for MN are (-1,3) and (7,10). If the ratio of the length of MX to the length of XN is 2:5, what is the y-coordinate of point X?
X is 2/7 of the way from M to N. So, X's y-value will be 2/7 of the distance from 3 to 10.
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To find the y-coordinate of point X, we first need to find its coordinates.
The length of segment MN can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of points M(-1, 3) and N(7, 10):
d = sqrt((7 - (-1))^2 + (10 - 3)^2)
= sqrt((8)^2 + (7)^2)
= sqrt(64 + 49)
= sqrt(113)
Let's assume the length of MX is 2k, and the length of XN is 5k (as given in the ratio).
Since X lies on the line segment MN, the sum of lengths MX and XN should be equal to the length of MN.
2k + 5k = sqrt(113)
7k = sqrt(113)
k = sqrt(113) / 7
Now, we can find the y-coordinate of point X by multiplying the length of MX (2k) by the slope of the line segment MN and adding it to the y-coordinate of point M (3).
The slope of line segment MN (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
m = (10 - 3) / (7 - (-1))
m = 7 / 8
Thus, the y-coordinate of point X is given by:
y-coordinate of point X = y-coordinate of point M + (length of MX * slope of MN)
= 3 + (2k * 7/8)
= 3 + (2 * (sqrt(113) / 7) * 7/8)
= 3 + (2 * sqrt(113) / 8)
= 3 + sqrt(113) / 4
Therefore, the y-coordinate of point X is 3 + sqrt(113) / 4.
To find the y-coordinate of point X, we first need to determine the coordinates of point M, point N, and point X.
Point M has the coordinates (-1, 3) and point N has the coordinates (7, 10).
Let's begin by finding the coordinates of point X.
To find the coordinates of point X, we need to use the ratio of the lengths of MX and XN (which is 2:5) to determine how far point X is along the line segment MN.
Let's assume that point X is located x units away from point M and y units away from point N.
The lengths of MX and XN can be determined using the distance formula:
Distance between two points (x₁, y₁) and (x₂, y₂) = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
Applying this formula, we can find the lengths of MX and XN:
MX = sqrt((x - (-1))² + (y - 3)²)
XN = sqrt((7 - x)² + (10 - y)²)
According to the ratio given, MX:XN = 2:5. We can set up the following equation based on this ratio:
MX / XN = 2 / 5
Substituting the distance values we found earlier:
sqrt((x - (-1))² + (y - 3)²) / sqrt((7 - x)² + (10 - y)²) = 2 / 5
To simplify the equation, let's square both sides:
[(x - (-1))² + (y - 3)²] / [(7 - x)² + (10 - y)²] = 4 / 25
Cross-multiply to get rid of the square roots:
25[(x - (-1))² + (y - 3)²] = 4[(7 - x)² + (10 - y)²]
Simplify the equation:
25(x² + 2x + 1 + y² - 6y + 9) = 4(x² - 14x + 49 + y² - 20y + 100)
Expand and collect like terms:
25x² + 50x + 25y² - 150y + 225 = 4x² - 56x + 196 + 4y² - 80y + 400
Simplify further:
25x² - 4x² + 50x + 56x + 25y² - 4y² - 150y + 80y = 196 + 400 - 225
Combine like terms:
21x² + 106x + 21y² - 70y = 371
At this point, we have an equation involving x and y coordinates. To find the y-coordinate of point X, we need to substitute the x-coordinate of point X. However, this information is not provided in the question. If you have the x-coordinate of point X, substitute it into the equation to find the corresponding y-coordinate.