In right triangle ABC, we have angle BAC = 90° and D is the midpoint of line AC. If AB = 7 and BC = 25, then what is tan of angle BDC?
Use Pythagoras theorem:
\(AC=\sqrt{25^2-7^2}=24\)
\(CD=DA=12\)(mid point)
Use Pythagoras theorem again:
\(BD=\sqrt{7^2+12^2}=\sqrt{193}\)
Find the sin of angle BCD:
\(\sin \angle BCD =\sin \angle BCA = \dfrac{7}{25}\)(angle BCD and angle BCA refers to the same angle)
Use law of sines on triangle BCD:
\(\dfrac{BD}{\sin\angle BCD}=\dfrac{BC}{\sin\angle BDC}\\ \dfrac{\sqrt{193}}{\frac{7}{25}}=\dfrac{25}{\sin\angle BDC}\\ \sin\angle BDC = \dfrac{25\times \frac{7}{25}}{\sqrt{193}}=\dfrac{7}{\sqrt{193}}\)
Construct a right-angled triangle with
legs = 7, -12(<-- negative because the angle is in the 2nd quadrant), hypotenuse=sqrt193
\(\tan \angle BDC =-\dfrac{7}{12}\)
Quicker way:
angle BDC = 90 deg + angle DBA
\(\boxed{\tan(a+b)=\dfrac{\sin(a+b)}{\cos(a+b)}=\dfrac{\sin a \cos b+\cos a \sin b}{\cos a \cos b - \sin a \sin b}=\dfrac{\tan a + \tan b}{1-\tan a \tan b}}\)
But because tan 90 deg does not have a meaning, we use the old definition:
\(\tan 90^{\circ}=\dfrac{1}{0}\)
So
\(\dfrac{1}{\tan 90^{\circ}}=0\)
\(\tan \angle BDC \\= \tan (90^{\circ}+\angle DBA)\\ =\dfrac{\tan 90^{\circ}+\tan{\angle DBA}}{1-\tan90^{\circ}\tan\angle DBA}\\ =\dfrac{1+\frac{\tan \angle DBA}{\tan 90^{\circ}}}{\dfrac{1}{\tan 90^{\circ}}-\tan \angle DBA}\\ =\dfrac{1+0\tan\angle DBA}{0-\tan\angle DBA}\\ =-\dfrac{1}{\tan\angle DBA}\)
AC = 24 using Pyth. thm. <-- I am lazy in typing
AD = 24/2 = 12
So tan angle DBA = 12/7
So tan angle BDC = -1/tan angle DBA = -7/12 <-- done :)