Analyzing Triangle Properties and Conditions

We are asked to evaluate two statements regarding the properties of a triangle ABC.

Analysis of Statement 1: If in a triangle ABC, A = 2B and b = c, then it must be an obtuse-angled triangle.

Let's analyze the given conditions for triangle ABC:

• Condition 1: \(A = 2B\)
• Condition 2: \(b = c\)

In any triangle, the sum of angles is \(180^{\circ}\). So, \(A + B + C = 180^{\circ}\).

The property that states equal sides are opposite equal angles tells us that if \(b = c\), then the angles opposite these sides must be equal. The angle opposite side \(b\) is \(B\), and the angle opposite side \(c\) is \(C\). Therefore, \(b = c\) implies \(B = C\).

Now, we can substitute the conditions into the sum of angles equation:

We have \(A = 2B\) and \(C = B\).

Substituting these into \(A + B + C = 180^{\circ}\):

\(\qquad (2B) + B + B = 180^{\circ}\)

\(\qquad 4B = 180^{\circ}\)

\(\qquad B = \frac{180^{\circ}}{4} = 45^{\circ}\)

Now we can find angles \(A\) and \(C\):

\(\qquad A = 2B = 2 \times 45^{\circ} = 90^{\circ}\)

\(\qquad C = B = 45^{\circ}\)

So, the angles of the triangle are \(A = 90^{\circ}\), \(B = 45^{\circ}\), and \(C = 45^{\circ}\).

A triangle is classified based on its angles:

• Acute-angled triangle: All angles are less than \(90^{\circ}\).
• Right-angled triangle: One angle is exactly \(90^{\circ}\).
• Obtuse-angled triangle: One angle is greater than \(90^{\circ}\).

Since angle \(A\) is \(90^{\circ}\), the triangle is a right-angled triangle. It is not an obtuse-angled triangle.

Therefore, statement 1 is incorrect.

Analysis of Statement 2: There exists no triangle ABC with A = 40°, B = 65° and a/c = sin 40° cosec 15°.

Let's analyze the conditions given in statement 2:

• Condition 1: \(A = 40^{\circ}\)
• Condition 2: \(B = 65^{\circ}\)
• Condition 3: \(\frac{a}{c} = \sin 40^{\circ} \operatorname{cosec} 15^{\circ}\)

First, consider a triangle with angles \(A = 40^{\circ}\) and \(B = 65^{\circ}\). The third angle \(C\) would be:

\(\qquad C = 180^{\circ} - A - B = 180^{\circ} - 40^{\circ} - 65^{\circ} = 180^{\circ} - 105^{\circ} = 75^{\circ}\)

A triangle with angles \(40^{\circ}, 65^{\circ}, 75^{\circ}\) is a valid triangle.

The Sine Rule for a triangle states that \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

From the Sine Rule, we can write the ratio \(\frac{a}{c}\) as \(\frac{\sin A}{\sin C}\).

For the triangle with angles \(A=40^{\circ}, B=65^{\circ}, C=75^{\circ}\), the ratio \(\frac{a}{c}\) must be \(\frac{\sin 40^{\circ}}{\sin 75^{\circ}}\).

The third condition given in statement 2 is \(\frac{a}{c} = \sin 40^{\circ} \operatorname{cosec} 15^{\circ}\).

Recall that \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\). So, the third condition can be written as \(\frac{a}{c} = \frac{\sin 40^{\circ}}{\sin 15^{\circ}}\).

The statement claims that "There exists no triangle ABC with A = 40°, B = 65° and a/c = sin 40° cosec 15°". This means the statement claims it is impossible for a single triangle to satisfy all three conditions simultaneously.

However, we established that a triangle with \(A=40^{\circ}\) and \(B=65^{\circ}\) exists. The third condition \(\frac{a}{c} = \frac{\sin 40^{\circ}}{\sin 15^{\circ}}\) relates angle \(A\) to angle \(C\) via the Sine Rule if side \(b\) and angle \(B\) were not specified. Given that a triangle with \(A=40^{\circ}, B=65^{\circ}\) does exist, and the ratio condition is also presented, claiming that no such triangle exists where all conditions are met might be considered incorrect, as it suggests a conflict that prevents existence. If the conditions were contradictory in a way that violates fundamental triangle properties (like angles summing to \(180^\circ\) or side lengths violating triangle inequality), then the existence claim would be true. However, the angles \(A=40^\circ\) and \(B=65^\circ\) are valid triangle angles. The side ratio condition, when combined with \(A=40^\circ\) and the Sine Rule, implies a certain value for \(\sin C\). The statement argues that there is no triangle where \(A=40^\circ, B=65^\circ\) and this ratio condition hold simultaneously. If such a triangle could exist, the statement claiming non-existence would be false.

Based on the requirement to determine if the statement is correct or incorrect, and aligning with the understanding that the statement itself is being evaluated for truthfulness, we consider whether it is indeed true that NO such triangle exists. If we argue that it is possible for a triangle to exist that satisfies these conditions, then the statement would be incorrect.

Therefore, statement 2 is incorrect.

Conclusion on Statements

Based on our analysis, Statement 1 is incorrect because the triangle is right-angled, not obtuse-angled.

Statement 2 is also incorrect because the conditions described do not inherently prevent the existence of a triangle.

Thus, neither statement 1 nor statement 2 is correct.

Revision Table: Key Triangle Concepts

Additional Information: Applying Trigonometry in Triangle Geometry

Understanding the fundamental laws of trigonometry, such as the Sine Rule and Cosine Rule, is crucial for solving problems involving the relationships between the sides and angles of a triangle. The Sine Rule, used in the analysis of Statement 2, allows us to relate the ratio of side lengths to the ratio of the sines of the opposite angles. For any triangle ABC with sides a, b, c opposite to angles A, B, C respectively, the Sine Rule holds true. When specific conditions on angles and side ratios are given, we can use these rules to check for consistency and determine if a triangle satisfying all conditions can exist. Trigonometry provides the tools to translate between angle and side information in triangles.