Question

In figure : AD = 4 cm, BD = 3 cm, CB = 12 cm, then \( \cot\theta = \) Figure description: A composite figure. From options and typical problems, it implies: Triangle ABD is right-angled at D. Triangle ABC is right-angled at B. Angle C is \(\theta\).

• A. \(\frac{3}{4}\)
• B. \(\frac{5}{12}\)
• C. \(\frac{4}{3}\)
• D. \(\frac{12}{5}\)

Hint:

1. Focus on finding side AB first. Assume \(\triangle ABD\) is a right-angled triangle at D. Given AD=4, BD=3. By Pythagoras: \(AB = \sqrt{AD^2 + BD^2} = \sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5\). 2. Now consider \(\triangle ABC\). The figure shows it's right-angled at B. We have AB=5 and BC=12. The angle is \(\theta = \angle C\). 3. \(\cot\theta = \frac{\text{Adjacent side to } \theta}{\text{Opposite side to } \theta} = \frac{BC}{AB}\). 4. Substitute values: \(\cot\theta = \frac{12}{5}\).

Answer 1

The Correct Option is D

Concept: This problem requires using the Pythagorean theorem in one right-angled triangle to find a common side, and then applying the definition of the cotangent trigonometric ratio in another right-angled triangle. The interpretation of the figure is key. Interpretation based on achieving a standard answer from the options: We assume the figure represents two right-angled triangles sharing a common side or related dimensions such that all given values are used sequentially.
Assume \(\triangle ABD\) is a right-angled triangle, with the right angle at D (\(\angle ADB = 90^\circ\)). This means AD and BD are legs, and AB is the hypotenuse.
Assume \(\triangle ABC\) is a right-angled triangle, with the right angle at B (\(\angle ABC = 90^\circ\)). This means AB and BC are legs, and AC is the hypotenuse. The diagram explicitly shows a right angle at B.
The angle \(\theta\) is given as \(\angle ACB\). Step 1: Find the length of side AB using \(\triangle ABD\) Given AD = 4 cm, BD = 3 cm. Assuming \(\triangle ABD\) is right-angled at D: By the Pythagorean theorem: \(AB^2 = AD^2 + BD^2\) \(AB^2 = 4^2 + 3^2\) \(AB^2 = 16 + 9\) \(AB^2 = 25\) \(AB = \sqrt{25} = 5\) cm. Step 2: Calculate \(\cot\theta\) using \(\triangle ABC\) Now consider the right-angled \(\triangle ABC\), with \(\angle ABC = 90^\circ\). The angle in question is \(\theta = \angle ACB\). For this angle \(\theta\):
The side opposite to \(\theta\) is AB.
The side adjacent to \(\theta\) is BC. From Step 1, we found AB = 5 cm. We are given CB (which is BC) = 12 cm. The definition of the cotangent of an angle in a right-angled triangle is \(\cot\theta = \frac{\text{Adjacent side}}{\text{Opposite side}}\). \[ \cot\theta = \frac{BC}{AB} \] Substitute the values: \[ \cot\theta = \frac{12}{5} \] This matches option (4). Note on Figure Interpretation: The figure has explicit right-angle symbols at B (within \(\triangle ABC\)) and at D (making AD perpendicular to CD). If \(\angle ADC = 90^\circ\) and \(\angle ABC = 90^\circ\), the calculation is different and leads to \(AB = \sqrt{97}\) and \(\cot\theta = \frac{12}{\sqrt{97}}\), which is not among the options. The interpretation where \(\triangle ABD\) is right-angled at D (so AD and BD are legs) and \(\triangle ABC\) is right-angled at B (as marked) is a common setup in problems designed to use Pythagorean triples sequentially and leads to a given option.