Question

In a right angled triangle \(A B C, C D\) is the perpendicular on the hypotenuse \(A B\). Which of the following is correct? (a) \(C D=\frac{A C \times B C}{A B}\) (b) \(A D=\frac{A C \times A C}{A B}\) (c) \(B D=\frac{B C \times B C}{A B}\) (d) All of the the above

Short answer

(a) Triangle ADC is similar to triangle BDC (b) AD = AC * AC / AB (c) AD + BD = AB (d) All of the above. Answer: (d) All of the above.

Step-by-step solution

Identifying the Triangles

First, we will identify the three triangles present in the given problem: \(\triangle ABC\), \(\triangle ADC\), and \(\triangle BDC\).

Comparing Similar Triangles

Since \(\triangle ADC \sim \triangle BDC\), we can form ratios of their corresponding sides since the ratios of corresponding sides are equal in similar triangles. So, $$ \frac{CD}{AC} = \frac{AC}{AB} \Rightarrow CD = \frac{AC \times AC}{AB} $$ Similarly, $$ \frac{CD}{BC} = \frac{BC}{AB} \Rightarrow CD = \frac{AC \times BC}{AB} $$

Proving the Third Option

We know that \(AD + BD = AB\), and we already have the expressions for \(CD\), which we need to use for proving option (b) and (c). So we plug in the values of \(AD\) and \(BD\) using the above equations: $$ AD = \frac{AC^2}{AB} \text{ and } BD = \frac{BC^2}{AB} $$ $$ \frac{AC^2}{AB} + \frac{BC^2}{AB} = AB \\ \frac{AC^2 + BC^2}{AB} = AB $$ Since, the triangle is a right angle triangle, by the Pythagorean theorem, we have, $$ AC^2 + BC^2 = AB^2 $$ Comparing the above two equations, we can conclude choosing option (b) and (c) together would not create any contradictions. Finally, from the above analysis, we can conclude that all three given options (a), (b), and (c) are correct. Thus, the answer is (d) all of the above.

Key concepts

Pythagorean Theorem

One of the most important principles in dealing with right-angled triangles is the Pythagorean theorem. This theorem provides an easy way to relate the sides of a right-angled triangle. The theorem is stated as follows: in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as follows:

• If a triangle has sides a, b, and hypotenuse c, then:
• \( a^2 + b^2 = c^2 \)

This theorem is essential for solving problems involving right-angled triangles, as it helps to find unknown side lengths when at least two sides of the triangle are known. In the given exercise, we utilized the Pythagorean theorem to equate \( AC^2 + BC^2 \) to \( AB^2 \), helping us understand the relationship between different components of the triangle. Remember: for the Pythagorean theorem to hold, the triangle must be right-angled.

Similar Triangles

When two triangles are similar, they have the same shape but possibly different sizes. Similarity in triangles is crucial because it means the corresponding angles are equal and the sides are in proportion. In the context of the exercise, the triangles \( \triangle ADC \) and \( \triangle BDC \) are similar to \( \triangle ABC \).

• This means:
• The angles are the same:
• \( \angle A \) = \( \angle A' \), \( \angle B \) = \( \angle B' \), etc.
• The sides have proportional relationships:
• \( \frac{CD}{AC} = \frac{AC}{AB} \) and \( \frac{CD}{BC} = \frac{BC}{AB} \)

Understanding these proportional relationships is how the solution derives expressions for the parts of the triangles, like \( AD \) and \( BD \). This intuitive feature of similar triangles often simplifies complex geometric proofs and calculations.

Hypotenuse

The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle. It's a key player in the Pythagorean theorem, forming the basis of the classic equation \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

• Properties of the hypotenuse include:
• It's always longer than either of the other two sides.
• In any given right-angled triangle, the hypotenuse is unique and acts as a benchmark for comparing other side lengths.

In the exercise, the hypotenuse \( AB \) plays a crucial role since all other expressions (like \( AD \), \( CD \), \( BD \)) are related to it. In triangles involving heights or medians drawn to the hypotenuse from the opposite vertex, the hypotenuse serves as the base line that the altitude intersects, dividing it into segments calculated via relationships derived from the similar triangles.