Question

In a triangle \(\mathrm{ABC}, \mathrm{D}\) is the mid point of \(\mathrm{BC}\). If \(\mathrm{AD}\) is perpendicular \(\mathrm{AC}\), then (a) \(\tan \mathrm{A}+\tan \mathrm{C}=0\) (b) \(\tan \mathrm{A}-\tan \mathrm{C}=0\) (c) \(\tan \mathrm{A}+2 \tan \mathrm{C}=0\) (d) \(2 \tan \mathrm{A}=\tan \mathrm{C}\)

Short answer

Answer: The correct relationship between the tangent of angle A and the tangent of angle C is \(2 \tan A = \tan C\).

Step-by-step solution

Draw the triangle

Draw triangle ABC with point D as the midpoint of BC. Since AD is perpendicular to AC, we have a right-angled triangle ACD and another right-angled triangle ADB.

Identify similar triangles

In triangle ACD and triangle ADB, we have: 1. Angle ADC = Angle ABD (both are right angles) 2. Angle DAC = Angle BAD (vertical opposite angles) Thus, we can say that triangle ACD is similar to triangle ADB (by AA similarity).

Write the proportionality of corresponding sides

Since the triangles are similar, the sides are proportional. Therefore, \(\frac{AD}{AC}=\frac{CD}{AD}\) (Ratio of corresponding sides in similar triangles ACD and ADB).

Find the tangent ratios of the angles A and C

Now, find the tangent ratios for each angle: \(\tan A = \frac{AD}{BD}\) \(\tan C = \frac{AD}{CD}\)

Relate the tangent ratios using the proportionality equation

Now substitute the tangent ratios into the proportionality equation from step 3: \(\frac{AD}{AC}=\frac{CD}{AD}\) We can rewrite this as: \(\frac{\tan A}{\tan C} = \frac{CD}{BD}\)

Simplify the relationship between tan A and tan C

Since D is the midpoint of BC, we have CD = BD, and hence: \(\frac{\tan A}{\tan C} = \frac{CD}{BD}\) Simplify this relationship to get the final answer: \(2 \tan A = \tan C\) The correct choice is (d) \(2 \tan \mathrm{A}=\tan \mathrm{C}\).

Key concepts

Midpoint Theorem

The Midpoint Theorem is a fundamental concept in geometry that often comes up in problems involving triangles. This theorem states that if you have a triangle \( \triangle ABC \), and you find the midpoint of one side, say \( BC \), and connect that midpoint to the opposite vertex \( A \), then the segment \( AD \) is both a median and a line that splits the triangle into two equal areas.
This line, \( AD \), is known as the median of \( \triangle ABC \), because it connects a vertex with the midpoint of the opposite side. The Midpoint Theorem can sometimes simplify proofs and calculations involving triangles and their properties. This theorem is useful because it helps establish a point that can lead to further significant geometrical insights about the triangle, as it did in the problem we are examining, where it contributed to finding a relationship between the tangents of angles \( \tan A \) and \( \tan C \).

Tangent in Triangles

The concept of tangent in triangles is crucial because it helps relate the angles to the sides in right-angled triangles. The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side. In our problem, this can be seen with \( \triangle ACD \) and \( \triangle ADB \), where \( \tan A = \frac{AD}{BD} \) and \( \tan C = \frac{AD}{CD} \).
The use of tangent is particularly handy in this problem as it allows us to form an equation relating \( \tan A \) and \( \tan C \), leading us to determine relationships such as \( 2 \tan A = \tan C \). Understanding how tangent ratios relate independent angles through shared sides can be a powerful tool in solving geometry problems, especially those involving trigonometric identities or simpler angle relationships.

Angle Bisector Theorem

The Angle Bisector Theorem is another key concept in geometry, especially when considering relationships inside triangles. It asserts that an angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. Although this theorem is not directly applied in the original exercise, understanding its principles can enhance your grasp of how different internal segments of a triangle behave.
By knowing how bisectors split sides in relation to other sides, you gain deeper insights into solving triangles when segments inside the triangle need to be evaluated. Even if not explicitly used in our solution, this theorem underpins many more complex geometric proofs and problems involving proportional relationships similar to those derived using the Midpoint Theorem.