Math Problem Statement

Point Q is on MN such that MQ : QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,5), determine the coordinates of Q.

Solution

The problem you've uploaded involves finding the coordinates of point Q on line segment MN, where the distance ratios are given as MQ:QN=2:3MQ : QN = 2 : 3. The coordinates of point M are (3,5)(3, 5), and point N has coordinates (8,5)(8, 5).

To solve this, we can use the section formula, which finds the coordinates of a point dividing a line segment in a given ratio.

The section formula states that the coordinates of a point QQ dividing a line segment joining M(x1,y1)M(x_1, y_1) and N(x2,y2)N(x_2, y_2) in the ratio m:nm : n are given by:

xQ=nx1+mx2m+nx_Q = \frac{n x_1 + m x_2}{m + n} yQ=ny1+my2m+ny_Q = \frac{n y_1 + m y_2}{m + n}

For your case:

  • M(3,5)M(3, 5) and N(8,5)N(8, 5),
  • The ratio MQ:QN=2:3MQ : QN = 2 : 3, so m=2m = 2 and n=3n = 3.

Now, applying the section formula:

For the xx-coordinate of Q: xQ=33+282+3=9+165=255=5x_Q = \frac{3 \cdot 3 + 2 \cdot 8}{2 + 3} = \frac{9 + 16}{5} = \frac{25}{5} = 5

For the yy-coordinate of Q: yQ=35+252+3=15+105=255=5y_Q = \frac{3 \cdot 5 + 2 \cdot 5}{2 + 3} = \frac{15 + 10}{5} = \frac{25}{5} = 5

Thus, the coordinates of point Q are (5,5)(5, 5).

Would you like more details on the section formula, or do you have any other questions related to this problem?

Further related questions:

  1. What would the coordinates of point Q be if the ratio was 3:2 instead of 2:3?
  2. How do you apply the section formula in 3-dimensional space?
  3. Can you find the midpoint of the line segment MN using the midpoint formula?
  4. What are the geometric interpretations of the ratios in the section formula?
  5. How do you apply the section formula when the line segment is vertical or horizontal?

Tip:

When solving problems involving line segments and ratios, always double-check the correct application of the section formula, as mixing up the coordinates or the ratio can lead to incorrect results!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Section Formula
Ratio and Proportion

Formulas

Section formula: (x_Q, y_Q) = ((n*x1 + m*x2) / (m+n), (n*y1 + m*y2) / (m+n))

Theorems

Section Formula in Coordinate Geometry

Suitable Grade Level

Grades 8-10