In what ratio does the point T(3, 0) divide the line segment joining S(4, 2) and U(1, 4) in the coordinate plane?

Introduction / Context:
This coordinate geometry question asks for the ratio in which a point divides a line segment joining two other points. Such problems are standard applications of the section formula or distance ratio concepts and are important for understanding how coordinates relate to geometric positions on the plane.

Given Data / Assumptions:

• Point S has coordinates (4, 2).• Point U has coordinates (1, 4).• Point T has coordinates (3, 0) and lies on the segment joining S and U.• We must find the ratio in which T divides the segment SU, that is, ST : TU.

Concept / Approach:
There are two common methods: using the section formula or using distances. If a point T divides the segment joining S(x1, y1) and U(x2, y2) in the ratio m : n internally, then its coordinates are given by ((m*x2 + n*x1) / (m + n), (m*y2 + n*y1) / (m + n)). Alternatively, the ratio of the distances ST and TU can be found directly using the distance formula between points. Here, it is simpler to calculate the distances ST and TU and then express their ratio in simplest form.

Step-by-Step Solution:
Step 1: Coordinates of S are (4, 2) and coordinates of T are (3, 0).Step 2: Use the distance formula to find ST: ST = sqrt((3 − 4)^2 + (0 − 2)^2).Step 3: Compute the differences: 3 − 4 = −1 and 0 − 2 = −2.Step 4: So ST = sqrt((−1)^2 + (−2)^2) = sqrt(1 + 4) = sqrt(5).Step 5: Now find TU using T(3, 0) and U(1, 4): TU = sqrt((1 − 3)^2 + (4 − 0)^2).Step 6: Compute the differences: 1 − 3 = −2 and 4 − 0 = 4.Step 7: So TU = sqrt((−2)^2 + (4)^2) = sqrt(4 + 16) = sqrt(20).Step 8: Simplify sqrt(20) = sqrt(4 * 5) = 2 * sqrt(5).Step 9: Now find the ratio ST : TU = sqrt(5) : 2*sqrt(5).Step 10: Divide both sides by sqrt(5) to simplify the ratio: 1 : 2.

Verification / Alternative check:
As an alternative check, the section formula can be used. Let T divide SU in the ratio k : 1, where k corresponds to the part near U and 1 is near S, or vice versa. However, once we have already computed ST : TU as 1 : 2, this automatically matches the idea that T is one third of the way from S to U. The fact that ST is exactly half of TU aligns with the distances we computed. The numeric distances and simplified ratio confirm that the ratio 1 : 2 is consistent and correct.

Why Other Options Are Wrong:
• 2 : 1: This would suggest that ST is twice TU, which is not true because ST = sqrt(5) and TU = 2*sqrt(5).• 2 : 3: This ratio does not match the actual distance comparison and would not place T at (3, 0).• 3 : 2: This would incorrectly suggest ST is 1.5 times TU, which contradicts the explicit distance calculations.

Common Pitfalls:
One common mistake is to confuse the order of the ratio and write TU : ST instead of ST : TU, which would invert the correct answer. Another error is to forget to take square roots properly or to simplify sqrt(20) incorrectly. Some learners also miscalculate coordinate differences, for example, mixing up x1 − x2 and x2 − x1, which leads to wrong distances. Being careful with arithmetic and the distance formula is crucial in such coordinate geometry problems.

Final Answer:
The point T divides the segment SU in the ratio 1 : 2 (that is, ST : TU = 1 : 2).