Math Problem Statement

The coordinates of the endpoints of QR are Q(-11, -12) and R(3, 2). Point S is on QR and divides it such that QS:RS is 5:2. What are the coordinates of S?

Solution

The problem asks for the coordinates of point SS, which divides line segment QRQR in the ratio QS:RS=5:2QS:RS = 5:2, given the coordinates of points Q(11,12)Q(-11, -12) and R(3,2)R(3, 2).

We can solve this using the section formula, which helps find a point dividing a line segment in a given ratio.

Section formula:

For a line segment with endpoints A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2), if a point PP divides the segment in the ratio m:nm:n, the coordinates of PP are:

P(mx2+nx1m+n,my2+ny1m+n)P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)

Here:

  • Q(11,12)Q(-11, -12) corresponds to (x1,y1)(x_1, y_1)
  • R(3,2)R(3, 2) corresponds to (x2,y2)(x_2, y_2)
  • The ratio is 5:25:2, so m=5m = 5 and n=2n = 2

Now let's calculate the coordinates of point SS. I'll do the math for you.The coordinates of point SS, which divides the line segment QRQR in the ratio 5:25:2, are (1.0,2.0)(-1.0, -2.0).

Would you like more details on the solution process or have any other questions?

Here are 5 related questions to expand on this topic:

  1. How do you find the midpoint of a line segment?
  2. Can the section formula be used for 3D coordinates?
  3. How do you handle the section formula when the ratio is negative?
  4. What is the geometric interpretation of dividing a line segment in different ratios?
  5. How would you solve if the ratio was reversed (i.e., QS:RS=2:5QS:RS = 2:5)?

Tip: Always double-check the ratio orientation (whether it's dividing from QQ to RR or the other way around) to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Ratios

Formulas

Section formula P((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

Theorems

Section formula for dividing a line segment

Suitable Grade Level

Grades 8-10