 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says check whether in the following AD is the bisector of angle A of triangle ABC given in figure 8.61 that is this figure. Now we have to tell whether AD is the bisector of this angle or not. So the measurements of the side given to us is first is AB is equal to 5 centimeter, AC is equal to 10 centimeter, BD equals to 1.5 centimeter and CD equals to 3.5 centimeter. The second part is AB is equal to 4 centimeter, AC is equal to 6 centimeter and BD equal to 1.6 centimeter and CD equals to 2.4 centimeter. The third part is AB is equal to 8 centimeter, AC equals to 24 centimeter and BD equal to 6 centimeter and BC equal to 24 centimeter. Now before starting with the solution let us see the key idea that we use in this question. The key idea behind this question is AD is the angle bisector of angle A in figure 8.61 that is this figure given to us only if AB by AC is equal to VD by CD that means we see that AD is the angle bisector of this angle only if AB by AC is equal to VD by CD. Now with this key idea in mind we'll start with the solution to this question. Let us start with the first part. Now here first of all we find AB by AC. Now it's given to us that AB is equal to 5 centimeter and AC is equal to 10 centimeter so AB by AC will be 5 by 10. Now if we divide the numerator and the denominator by 5 we get 1 by 2. Now we find BD by CD. Now BD is given to us as 1.5 centimeter and CD is given to us as 3.5 centimeter so BD over CD is 1.5 divided by 3.5 and that is equal to 3 by 7. Now AB over AC is not equal to BD over CD that is 1 by 2 is not equal to 3 by 7 hence AD is not the angle bisector of angle A. So for the first part we see that AD is not the angle bisector of this angle. Now let us see the solution to the second part. Again here we will find AB by AC. Now AB given to us is 4 centimeter and AC given to us is 6 centimeter so AB by AC is 4 by 6. If we divide the numerator denominator by 2 we get 2 by 3. Now again we find BD by CD. BD by CD will be 1.6 centimeter by 2.4 centimeter and that is equal to 2 by 3. Now because AB by AC is equal to BD by CD that is 2 by 3 is equal to 2 by 3 hence AD is the bisector of angle A. Now let us see the solution to the third part. Again we find AB by AC. Now AB by AC will be 8 centimeter by 24 centimeter so it is 8 by 24. Now if we divide the numerator and the denominator by 8 we get 1 by 3. Now we see that BC given to us is 24 centimeter that is this is 24 centimeter here we are given BD that is 6 centimeter. So we can find out CD. CD will be equal to BC minus BD. Now BC is given to us it is 24 centimeter and BD is given to us is 6 centimeter. So CD is 24 minus 6 that is 18 centimeter. Now we can find BC by now we can find BD by CD. So therefore BD by CD is equal to 6 by 18 again we divide the numerator denominator by 6 we get 1 by 3. So we see that AB by AC is equal to BD that is 1 by 3. So because AB by AC is equal to BD by CD hence we say that AD is the bisector of angle A and this is our answer to third part. So let us summarize our answer. Now to the first part we see that AD is not the angle bisector of angle A. So the first part answer is no to the second part and to the third part we see that AD is the angle bisector of angle A. So our answer is yes to the second part and yes to the third part. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.