Download App

Triangles — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTriangles2 Marks
Question
In a $\triangle\text{ABC},\text{AD}$ is a median and $\text{AL}\perp\text{BC}.$
Prove that:
  1. $\text{AC}^2=\text{AD}^2+\text{BC}.\text{DL}+\Big(\frac{\text{BC}}{2}\Big)^2$
  2. $\text{AB}^2=\text{AD}^2-\text{BC}.\text{DL}+\Big(\frac{\text{BC}}{2}\Big)^2$
  3. $\text{AC}^2+\text{AB}^2=2\text{AD}^2+\frac{\text{1}}{2}\text{BC}^2$
✓

Answer


In right $\triangle\text{ALD},$
By Pythagoras theorem,
$AD^2 = AL^2 + DL^2$
$\Rightarrow AL^2 = AD^2 - DL^2 ....(i)$
In right $\triangle\text{ACL},$
By Pythagoras theorem,
$\text{AC}^2 =\text{ AL}^2 + \text{LC}^2$
$\Rightarrow \text{AC}^2 = (\text{AD}^2 -\text{ DL}^2) + (\text{DL} +\text{ DC)}^2\dots(\text{from (i))}$
$\Rightarrow\text{AC}^2=\text{AD}^2-\text{DL}^2+\Big(\text{DL}+\frac{\text{BC}}{2}\Big)^2$.....(Since AD is the median)
$\Rightarrow\text{AC}^2=\text{AD}^2-\text{DL}^2+\text{DL}^2+\Big(\frac{\text{BC}}{2}\Big)^2+\text{BC}.\text{DL}$
$\Rightarrow\text{AC}^2=\text{AD}^2+\text{BC}.\text{DL}+\Big(\frac{\text{BC}}{2}\Big)^2$
Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from Triangles→Triangles — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

In FIg. check whether AD is the bisector of $\angle\text{A}$ of $\triangle\text{ABC}$ in the following.

AB = 5cm, AC = 12cm, BD = 2.5cm and BC = 9cm.
→
Write the discriminant of the following quadratic equation.
$x^2 - x + 1 = 0$
→
The median value for the following frequency distribution is $35$ and the sum of the all frequencies is $170$. Using the formula for median, find the missing frequencies.
Class
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Frequency
10
20
?
40
?
25
15
→
Evaluate the following:
$\frac{\sin19^\circ}{\cos71^\circ}$
→
Represent the situation in the form of the quadratic equation:
A train travels a distance of $480$ km at a uniform speed. If the speed had been 8 km/hr less, then it would have taken $3$ hours more to cover the same distance. We need to find the speed of the train.
→
If A, B, C are the interior angles of a triangle ABC, prove that.
$\sin\Big(\frac{\text{B+C}}{2}\Big)=\cos\frac{\text{ A}}{2}$
→
Find the value of x in the following:
$\sqrt{3}\sin\text{x}=\cos\text{x}$
→
Find the LCM and HCF of 8, 9 and 25 integers by applying the prime factorisation method.
→
A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
Neither an ace nor a king.
→
Show that $(\sqrt{3}+\sqrt{5})^2$ is an irrational number.
→