In the right triangle right angled at sqrt 31 and Express ta...

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--- ## Question 1 **Question content**: In the right triangle \( BCD \) (right - angled at \( C \)), \( CD = \sqrt{31} \) and \( BC = 12 \). Express \( \tan B \) as a fraction in simplest terms. **Discipline**: Mathematics **Grade**: High School (usually covered in trigonometry, which is typically introduced in high school, e.g., 9th - 10th grade) ### Correct answer \(\boldsymbol{\dfrac{\sqrt{31}}{12}}\) ### Detailed problem - solving steps 1. **Recall the definition of tangent in a right triangle**: In a right triangle, the tangent of an acute angle is defined as the ratio of the length of the side **opposite** the angle to the length of the side **adjacent** to the angle. Mathematically, for an acute angle \( \theta \) in a right triangle, \( \tan\theta=\dfrac{\text{length of opposite side}}{\text{length of adjacent side}} \). 2. **Identify the sides relative to \( \angle B \)**: - In right triangle \( BCD \) (right - angled at \( C \)), for \( \angle B \): - The side **opposite** \( \angle B \) is \( CD \), and its length is \( \sqrt{31} \). - The side **adjacent** to \( \angle B \) is \( BC \), and its length is \( 12 \). 3. **Calculate \( \tan B \)**: Using the definition of tangent, we substitute the lengths of the opposite and adjacent sides into the formula. So, \( \tan B=\dfrac{\text{opposite to } \angle B}{\text{adjacent to } \angle B}=\dfrac{CD}{BC}=\dfrac{\sqrt{31}}{12} \). ### Knowledge points involved 1. **Right triangle trigonometry (tangent of an acute angle)**: - **Detailed interpretation**: In a right triangle, for any acute angle \( \theta \), the tangent of \( \theta \) (denoted as \( \tan\theta \)) is the ratio of the length of the side opposite \( \theta \) to the length of the side adjacent to \( \theta \). That is, \( \tan\theta=\dfrac{\text{opposite}}{\text{adjacent}} \). This concept is fundamental in trigonometry and is used to relate the angles of a right triangle to the lengths of its sides, which is useful in solving problems related to heights, distances, and various geometric applications. 2. **Identification of opposite and adjacent sides in a right triangle**: - **Detailed interpretation**: When dealing with an acute angle in a right triangle, the "opposite" side is the side that does not form the angle (it is across from the angle), and the "adjacent" side is the side that forms the angle along with the hypotenuse (it is one of the two legs that is part of the angle). Properly identifying these sides is crucial for correctly applying trigonometric ratios like tangent, sine, and cosine. 3. **Simplification of fractions (with radicals)**: - **Detailed interpretation**: A fraction is in simplest form when the numerator and denominator have no common factors other than 1. In the case of a fraction with a radical in the numerator (like \( \dfrac{\sqrt{31}}{12} \)), since \( \sqrt{31} \) is an irrational number and 31 is a prime number, \( \sqrt{31} \) and 12 have no common factors (other than 1), so the fraction is already in its simplest form. This concept is important for presenting the final answer in a standard, simplified manner. ---

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