MCQ
In an equilateral triangle $ABC$ if $\text{AD}\perp\text{BC},$ then:
- A$5AB^2 = 4AD^2$
- ✓$3AB^2 = 4AD^2$
- C$4AB^2 = 3AD^2$
- D$2AB^2 = 3AD^2$
✓
Answer
Correct option: B.
$3AB^2 = 4AD^2$
$\triangle\text{ABC}$ is an equilateral triangle and $\text{AD}\perp\text{BC}.$
In $\triangle\text{ABD},$ applying Pythagoras theorem, we get,
$\text{AB}^2=\text{AD}^2+\text{BD}^2$
$\text{AB}^2=\text{AD}^2+\Big(\frac{1}{2}\text{BC}\Big)^2\Big(\because\text{BD}=\frac{1}{2}\text{BC}\Big)$
$\text{AB}^2=\text{AD}^2+\Big(\frac{1}{2}\text{AB}\Big)^2\Big(\because\text{AB}=\text{BC}\Big)$
$\text{AB}^2=\text{AD}^2+\frac{1}{4}\text{AB}^2$
$3\text{AB}^2=4\text{AD}^2$
We got the result as $B$.
In $\triangle\text{ABD},$ applying Pythagoras theorem, we get,
$\text{AB}^2=\text{AD}^2+\text{BD}^2$
$\text{AB}^2=\text{AD}^2+\Big(\frac{1}{2}\text{BC}\Big)^2\Big(\because\text{BD}=\frac{1}{2}\text{BC}\Big)$
$\text{AB}^2=\text{AD}^2+\Big(\frac{1}{2}\text{AB}\Big)^2\Big(\because\text{AB}=\text{BC}\Big)$
$\text{AB}^2=\text{AD}^2+\frac{1}{4}\text{AB}^2$
$3\text{AB}^2=4\text{AD}^2$
We got the result as $B$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Which of the following options shows the highest probability.
Out of the following which is the Pythagorean triplet?
If sin $\alpha$ and $\cos\alpha$ are the roots of the equations $ax^2 + bx + c = 0,$ then $b^2 =$
→$\left|\begin{array}{cc}3 & 5 \\ 2 & x\end{array}\right|=2 \quad \therefore x=\cdots \cdots$
→The formula to find mean by direct method is
Mark the correct alternative in the following:
If the sum of n terms of an $A.P.$ is $3n^2 + 5n$ then which of its terms is $164?$
→If the sum of n terms of an $A.P.$ is $3n^2 + 5n$ then which of its terms is $164?$
Cards, each marked with one of the numbers $6, 7, 8, ..., 15$ are placed in a box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting a card with number less than $10?$
→Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6,EB = 10, CE = 8, find ED.
For an A.P. if $t_4=12$ and $d=-10$ then find $a$.
→Volumes of two spheres are in the ratio $64 : 27$. The ratio of their surface areas is:
→