Question

The construction of a triangle \(\triangle \text{ABC}\) given that \(\text{BC} = 6 \text{ cm}\), \(\angle \text{B} = 45^\circ\) is not possible, when the difference of AB and AC is equal to :

• A. \(6.9 \text{ cm}\)
• B. \(5.2 \text{ cm}\)
• C. \(5.0 \text{ cm}\)
• D. \(4.0 \text{ cm}\)

Hint:

When constructing a triangle with a given side (e.g., BC), an adjacent angle (\(\angle \text{B}\)), and the difference of the other two sides (\(|AB - AC|\)): Remember this rule: The construction is {not possible} if the difference \(|AB - AC|\) is {greater than or equal to} the given side BC. Given \(BC = 6 \text{ cm}\). Look for an option where the difference \(\geq 6 \text{ cm}\). Option (1) is \(6.9 \text{ cm}\), and \(6.9>6\), so this is when construction is not possible.

Answer 1

The Correct Option is A

Concept: For the construction of a triangle \(\triangle \text{ABC}\) when one side (say, BC), an angle adjacent to it (\(\angle \text{B}\)), and the difference between the other two sides (\(|AB - AC|\)) are given, there's a specific condition for the construction to be possible. Step 1: Condition for Possibility/Non-Possibility of Construction When constructing a triangle given side BC (let its length be \(a\)), \(\angle \text{B}\), and the difference \(|AB - AC| = d\), the construction of a non-degenerate triangle is possible only if the difference \(d\) is less than the given side \(a\) (i.e., BC). \[ |AB - AC|<BC \] If the difference \(|AB - AC|\) is equal to or greater than BC, the construction of such a triangle is not possible. \[ |AB - AC| \geq BC \quad (\text{Construction not possible}) \] Step 2: Apply the given values to the condition In this problem:
Length of side BC = \(6 \text{ cm}\)
\(\angle \text{B} = 45^\circ\)
We need to find when the construction is {not possible} based on the difference \(|AB - AC|\). According to the condition, construction is not possible if: \[ |AB - AC| \geq 6 \text{ cm} \] Step 3: Evaluate the options We check each option to see if it satisfies the condition for non-possibility (\(|AB - AC| \geq 6 \text{ cm}\)):
Option (1): Difference = \(6.9 \text{ cm}\) Is \(6.9 \text{ cm} \geq 6 \text{ cm}\)? Yes, \(6.9>6\). Therefore, if the difference is \(6.9 \text{ cm}\), the construction is not possible.
Option (2): Difference = \(5.2 \text{ cm}\) Is \(5.2 \text{ cm} \geq 6 \text{ cm}\)? No, \(5.2<6\). Construction is possible.
Option (3): Difference = \(5.0 \text{ cm}\) Is \(5.0 \text{ cm} \geq 6 \text{ cm}\)? No, \(5.0<6\). Construction is possible.
Option (4): Difference = \(4.0 \text{ cm}\) Is \(4.0 \text{ cm} \geq 6 \text{ cm}\)? No, \(4.0<6\). Construction is possible. Step 4: Conclusion The construction of the triangle is not possible when the difference of AB and AC is \(6.9 \text{ cm}\), as this value is greater than the length of the side BC.