In the given figure if DE || BC, then K is given by

Appeared in: UPPSC 2023

Explanation

The problem is based on the property of similar triangles, which states that if two triangles are similar, the ratio of their corresponding sides is equal.
Since the line segment DE is parallel to the base BC, the smaller triangle (ADE) is similar to the larger triangle (ABC).
The ratio of the sides AD/AB is equal to the ratio of the bases DE/BC. With AD=3, AB=3+3=6, and DE=4, the equation is 3/6 = 4/K.
Solving the equation 1/2 = 4/K gives K = 8.

Option A: This is the length of the upper parallel line segment (DE), not the base (BC or K).
Option C: This is the total length of the side AB (AD + DB = 3 + 3 = 6). It incorrectly equates the length of the base K to the length of the side AB.
Option D: This value may be obtained by an incorrect calculation, such as subtracting DE from AB (6 - 4 = 2) or by incorrectly setting up the ratio.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Properties of similar triangles and the Basic Proportionality Theorem (Thales's Theorem) as background academic context rather than a clinical decision trigger.
This question tests fundamental mathematical and logical reasoning skills, which are essential for various calculations in nursing practice, such as drug dosage, IV drip rates, and data interpretation.
Understanding proportions is crucial for scaling dosages based on patient weight or for diluting solutions correctly.
What if? If point D was not the midpoint (e.g., AD=2 and DB=4), the ratio would change. The calculation would be 2/(2+4) = 4/K, which is 2/6 = 4/K or 1/3 = 4/K, leading to K=12. This highlights the importance of using the correct proportions.

How to Approach the Question

First, analyze the given image and identify the geometric figure, which is a triangle with a line parallel to its base.
Recognize the key information provided: DE is parallel to BC (DE || BC). This is the crucial clue that indicates the triangles ADE and ABC are similar.
Recall the property of similar triangles: the ratios of their corresponding sides are equal.
Set up the proportion using the corresponding sides: AD/AB = DE/BC.
Substitute the known values from the figure: AD=3, AB = AD+DB = 3+3=6, DE=4, and BC=K.
Solve the resulting equation (3/6 = 4/K) for the unknown variable K.

Concept Tested & Keywords

Concept Tested: Properties of similar triangles and the Basic Proportionality Theorem (Thales's Theorem).
Stem keywords: triangle, DE || BC, figure
Lead-in keywords: K is given by

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