Let n be a positive integer. In triangle ABC, AB = 3n, AC = 2n + 15, BC = n + 30, and angle \(A > \angle B > \angle C\). How many possible values of n are there?

envoy May 22, 2020

#1**-2 **

The possible values of n are 8, 9, 10, and 11, giving us 4 possible values of n.

Guest May 23, 2020

#2**+1 **

There are more than 4 values.

1) First draw up a triangle that looks possible.

All the angles are different so if you use a triangle that looks a bit like a 30,60, 90 degree triangle that will be good to work with.

2) Label the 3 angles correctly according to their relative angle size

3) Label the 3 sides

4) Now the smallest side is opposite the smallest angle etc so write 'smallest' and 'biggest' near the correct sides.

5) Now you know that smallest side < middle side Solve the inequality

6) You also know that the middle side < longest side. Solve the inequality.

7) So what do you know so far?

8) Now to be a triangle at all the sum of the 2 smallest sides must be longer than the long side.

Form the inequality and solve it.

9) So what have you found.

If you have questions then ask. You can take a photo of your pic on your phone and post it if need be.

I am very happy to help more if necessary but i want to be convinced that you are trying to help yourself first.

**Please no one else answer**

Melody May 24, 2020

#3**+2 **

Thank you for you help!

Basically what I did was I made all the inequalities.

I got

3n+2n+15>n+30 --------> n>15/2

3n+n+30> 2n+15 -------> n>15/4

2n+15+n+30>3n (no use)

I had to make a diagram for this next part.

We know BC>AC because largest side is opposite largest angle, smallest side is opposite smallest angle. We also know AC>AB from similar reasoning. This means

n+30>2n+15 -------> n<15

2n+15>3n ------> n<15

There are 11 values of n that satisfy all of the inequalities

envoy
May 24, 2020