Question
In $\triangle\text{ABC},\ \angle\text{A}=60^\circ.$ Prove that $BC^2 = AB^2 + AC^2 - AB \times AC.$
✓
Answer
In $\triangle\text{ABC}$ in which A is an acute angle with $60^\circ.$
$\sin60^\circ=\frac{\text{CD}}{\text{AC}}=\frac{\sqrt{3}}{2}$
$\Rightarrow\text{CD}=\frac{\sqrt{3}}{2}\text{AC}\ \ \ ....(1)$
$\cos60^\circ=\frac{\text{AD}}{\text{AC}}=\frac{1}{2}$
$\Rightarrow\text{AD}=\frac{1}{2}\text{AC}\ \ \ ....(2)$
Now apply Pythagoras theorem in triangle BCD
$BC^2 = CD^2 + BD^2$
$= CD^2 + (AB - AD)^2$
$=\Big(\frac{\sqrt{3}}{2}\text{AC}\Big)+\text{AB}^2+\Big(\frac{1}{2}\text{AC}\Big)^2-2\text{AB}\frac{1}{2}\text{AC}$
$= AC^2 + AB^2 - AB \times AC$
Hence $BC^2 = AB^2 + AC^2 - AB \times AC$
$\sin60^\circ=\frac{\text{CD}}{\text{AC}}=\frac{\sqrt{3}}{2}$
$\Rightarrow\text{CD}=\frac{\sqrt{3}}{2}\text{AC}\ \ \ ....(1)$
$\cos60^\circ=\frac{\text{AD}}{\text{AC}}=\frac{1}{2}$
$\Rightarrow\text{AD}=\frac{1}{2}\text{AC}\ \ \ ....(2)$
Now apply Pythagoras theorem in triangle BCD
$BC^2 = CD^2 + BD^2$
$= CD^2 + (AB - AD)^2$
$=\Big(\frac{\sqrt{3}}{2}\text{AC}\Big)+\text{AB}^2+\Big(\frac{1}{2}\text{AC}\Big)^2-2\text{AB}\frac{1}{2}\text{AC}$
$= AC^2 + AB^2 - AB \times AC$
Hence $BC^2 = AB^2 + AC^2 - AB \times AC$
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
Prove that the points $(7,10),(-2,5)$ and $(3,-4)$ are the vertices of an isosceles right triangle.
→If the volumes of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then write the ratio of their height.
A person rowing at the rate of 5km/ h in still water, takes thrice as much time in going 40km upstream as in going 40km downstream. Find the speed of the stream.
A canal is 300cm wide and 120cm deep. The water in the canal is flowing with a speed of 20km/h. How much area will it irrigate in 20 minutes if 8cm of standing water is desired?
Find the number of revolutions made by a circular wheel of area $1.54\ m^2$ in rolling a distance of $17\ m$.
→A chord PQ of a circle of radius 10cm subtends an angle of $60^\circ$ at the centre of the circle. Find the area of major and minor segments of the circle. $\Big[\text{Use }\pi=\frac{22}{7}\Big]$
→A girl of heigh 90cm is walking away from the base of a lamp-post at a speed of 1.2m/sec. If the lamp is 3.6m above the ground, find the length of her shadow after 4 seconds.
Find the volume of a solid in the form of a right circular cylinder with hemi-spherical ends whose total length is 2.7m and the diameter of each hemi-spherical end is 0.7m.
If the mid-point of the segment joining $A(x, y+1)$ and $B(x+1, y+2)$ is $C\left(\frac{3}{2}, \frac{5}{2}\right)$, find $x, y$.
→Prove the following trigonometric identities.
$\frac{1+\sin\theta}{\cos\theta}+\frac{\cos\theta}{1+\sin\theta}=2\sec\theta$
→$\frac{1+\sin\theta}{\cos\theta}+\frac{\cos\theta}{1+\sin\theta}=2\sec\theta$