Question

In $\triangle ABC$, $\angle B$ is a right angle, $AC = 6$ cm, and $D$ is the mid-point of $AC$. The length of $BD$ is

• A. 4 cm
• B. $\sqrt{6}$ cm
• C. 3 cm
• D. 3.5 cm

Hint:

Median to hypotenuse in a right triangle is half the hypotenuse.

Answer 1

The Correct Option is C

In the given right-angled triangle $\triangle ABC$, where $\angle B = 90^\circ$, let's determine the length of $BD$, with $D$ being the midpoint of $AC$. Given that $AC = 6$ cm, we have the following: 

1. Since $D$ is the midpoint of $AC$, the length of $AD = DC = \frac{AC}{2} = \frac{6}{2} = 3$ cm.
2. In $\triangle ABC$, since $\angle B = 90^\circ$, by the Pythagorean theorem, $AB^2 + BC^2 = AC^2 = 6^2 = 36$. However, we need $BD$ which involves another approach.
3. Now, consider the right triangle formed by $\triangle ABD$. As $D$ is the midpoint of $AC$, $BD$ can be determined using the median theorem (Apollonius's theorem):
   $BD^2 = \frac{2AB^2 + 2AD^2 - AC^2}{4}$. 
   Given that $AD = 3$ cm, substitute in the expression:
   As $BD$ is the median and $D$ is the midpoint, $\frac{2AB^2 + 2 \times 3^2 - 6^2}{4} = \frac{2AB^2 + 18 - 36}{4} = \frac{2AB^2 - 18}{4}$.
4. From the geometry of the triangle, the length of $BD$ in a right triangle where one side is a median is always $=\frac{1}{2}\times \text{hypotenuse}$ if $D$ lies on hypotenuse $AC$. Therefore, $BD = 3$ cm.