GATE2025_CS1_Q7

In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?

Note: The figure shown is representative.

✅ Final Answer:

The correct answer is option (A) 1/3.

🔹 Short Answer:

Given that the height of trapezium SQRT is half the total height of ΔPQR, the height of the smaller triangle ΔPST must also be half the total height. Since ΔPST is similar to ΔPQR, the ratio of their areas is the square of the ratio of their heights: (1/2)² = 1/4. If Area(ΔPQR) = 4 units, then Area(ΔPST) = 1 unit. The Area(Trapezium SQRT) is the difference, 4 - 1 = 3 units. The required ratio of Area(ΔPST) to Area(Trapezium SQRT) is 1/3.

🔸 Long Answer:

This geometry problem is solved using the properties of similar triangles. Let's break it down step-by-step.

Step 1: Understand the Geometry and Given Information

• Since line ST is parallel to line QR (ST || QR), by the property of corresponding angles, triangle PST is similar to triangle PQR (ΔPST ~ ΔPQR).
• Let H be the total height (altitude) of ΔPQR from point P to the base QR.
• Let h be the height of ΔPST from point P to the base ST.
• The height of the trapezium SQRT is the perpendicular distance between the parallel lines ST and QR, which is H - h.
• The problem states: Height of Trapezium = (1/2) * Total Height.

Step 2: Formulate and Solve for the Heights

From the given information, we can write the equation:

H - h = H / 2

Solving for h, we get:

h = H - H / 2 = H / 2

This is a key finding: the height of the smaller triangle (ΔPST) is exactly half the height of the larger triangle (ΔPQR).

Step 3: Apply the Area Ratio Theorem for Similar Triangles

A fundamental theorem in geometry states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding heights (or sides).

Area(ΔPST) / Area(ΔPQR) = (h / H)²

Substituting our result from Step 2 (h = H/2):

Area(ΔPST) / Area(ΔPQR) = ((H/2) / H)² = (1/2)² = 1/4

Step 4: Calculate the Area of the Trapezium and the Final Ratio

The area of the trapezium SQRT is the area of the large triangle minus the area of the small triangle.

Area(Trapezium SQRT) = Area(ΔPQR) - Area(ΔPST)

From Step 3, we know that Area(ΔPST) = (1/4) * Area(ΔPQR). Substituting this into the equation above:

Area(Trapezium SQRT) = Area(ΔPQR) - (1/4) * Area(ΔPQR) = (3/4) * Area(ΔPQR)

Now, we can find the required ratio:

Ratio = Area(ΔPST) / Area(Trapezium SQRT)
Ratio = [ (1/4) * Area(ΔPQR) ] / [ (3/4) * Area(ΔPQR) ]
Ratio = (1/4) / (3/4) = 1/3

Thus, the ratio of the area of the triangle PST to the area of the trapezium SQRT is 1/3.

Bibliography:
- NCERT. (2022). Mathematics Textbook for Class X. National Council of Educational Research and Training. (Chapter 6: Triangles).
- Hall, H. S., & Stevens, F. H. (1893). A Text-Book of Euclid's Elements. Macmillan and Co.

Ask your doubts in comment form our GURUJEE will reply ASAP, click notify me button so that you get notified on reply.