Question
In a $\triangle\text{ABC}, AB = BC = CA = 2a$ and $\text{AD}\perp\text{BC}.$ Prove that
$\text{AD}=\text{a}\sqrt{3}$
$\text{AD}=\text{a}\sqrt{3}$
✓
Answer
$\triangle\text{ABC},$ AB = BC = AC = 2a
$\triangle\text{AD}\perp\text{BC}$
AD bisects BC at D
$\text{BD}=\text{DC}=\frac{1}{2}\text{BC}=\text{a}$
Now in $\triangle\text{ABD}$
$AB^2 = AD^2 + BD^2$
$\Rightarrow (2a)^2 = AD^2 + a^2 \Rightarrow 4a^2 = AD^2 + a^2$
$\Rightarrow AD^2 = 4a^2 - a^2 = 3a^2$ $=(\sqrt3\text{a})^2$
$\therefore\text{AD}=\sqrt{3}\text{a}$
$\triangle\text{AD}\perp\text{BC}$
AD bisects BC at D
$\text{BD}=\text{DC}=\frac{1}{2}\text{BC}=\text{a}$
Now in $\triangle\text{ABD}$
$AB^2 = AD^2 + BD^2$
$\Rightarrow (2a)^2 = AD^2 + a^2 \Rightarrow 4a^2 = AD^2 + a^2$
$\Rightarrow AD^2 = 4a^2 - a^2 = 3a^2$ $=(\sqrt3\text{a})^2$
$\therefore\text{AD}=\sqrt{3}\text{a}$
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
In two concentric circles, a chord of length $8 \ cm$ of the larger circle touches the smaller circle. If the radius of the larger circle is $5 \ cm,$ then find the radius of the smaller circle.
→If $\sin\theta=\frac{1}{3},$ then find is the value of $2\cot^2\theta+2$.
→In the given figure, $\triangle AHK \sim \Delta ABC$. If $AK =8 \ cm, BC =3.2 \ cm$ and $HK =6.4 \ cm$, then find the length of $AC .$
→Define HCF of two positive integers and find the HCF of the following pairs of numbers:
70 and 30
70 and 30
For the following distribution draw a 'less than type' ogive and from the curve find the median.
Mean, Median, Mode of grouped Data, Cumulative Frequency Graph and Ogive.
|
Marks obtained
|
Less than 20
|
Less than 30
|
Less than 40
|
Less than 50
|
Less than 60
|
Less than 70
|
Less than 80
|
Less than 90
|
Less than 100
|
|
Number of students
|
2
|
7
|
17
|
40
|
60
|
82
|
85
|
90
|
100
|
Very-Short and Short-Answer Questions:
A man has some hens and cows. If the number of heads be 48 and the number of feet be 140, how many cows are there?
A man has some hens and cows. If the number of heads be 48 and the number of feet be 140, how many cows are there?
Very-Short and Short-Answer Qustions:
If $\cos\theta=\frac23,$ write the value of $\big(4+4\tan^2\theta\big).$
→If $\cos\theta=\frac23,$ write the value of $\big(4+4\tan^2\theta\big).$
Write the area of the sector of a circle whose radius is r and length of the arc is l.
Find the coordinate of the points of trisection of the line segment joining the points A(7, -2) and B(1, -5).
Very-Short and Short-Answer Questions:
The sum of two numbers is 80. The larger number exceeds four times the smaller one by 5. Find the numbers.
The sum of two numbers is 80. The larger number exceeds four times the smaller one by 5. Find the numbers.