the sizes of the interior angles of a hexagon ABCDEF form an arithmetic sequence...

[bumblebee123]

Can anyone help to explain what to do? I'm really confused.

question: the sizes of the interior angles of a hexagon ABCDEF form an arithmetic sequence. the smallest interior angle of the hexagon is 75 degrees. find the size of the largest interior angle of the hexagon.

any help would be really appreciated, thanks!

 

[Dr.Peterson]

What do you know about the sum of the interior angles of any hexagon?

 

[HallsofIvy]

Here's one way of thinking about the interior angles of a polygon. Choose any point in the interior of the polygon and draw the line segments from that point to each vertex. That divides the polygon into n triangles where n is the number of sides of the polygon (for a hexagon, n= 6). Since the angles in any triangle sum to 180 degrees, the angles in those n triangles sum to 180n. But that includes the angles at that interior point. Since the those angle form a complete circle around the interior point they total 360 degrees. We need to subtract those off: the interior angles in a polygon with n sides total 180n- 360= 180(n- 2).

 

[bumblebee123]

What do you know about the sum of the interior angles of any hexagon?

The sum of the interior angles for any polygon = (2n - 4) x 90

therefore, the sum of the interior angles of a hexagon = ( 12 - 4 ) x 90 = 8 x 90 = 720 ( degrees )

 

[Dr.Peterson]

So now you know that the first term of the sequence is 75 and the sum of six terms is 720. You know where to go from here, right? Give it a try.

(You don't have to stop at answering our direct questions; they are meant to prod you in the right direction so you can continue.)

 

[bumblebee123]

So now you know that the first term of the sequence is 75 and the sum of six terms is 720. You know where to go from here, right? Give it a try.

(You don't have to stop at answering our direct questions; they are meant to prod you in the right direction so you can continue.)

oh, okay! thanks, I'll try.

so 720 = 6/2 ( (2x75) + (n-1)d )

720 = 3 (150 + ( 6-1 )d )

720 = 3 (150 + 5d )

720 = 450 + 15d

270 = 15d

d = 18

sorry, I really don't know what to do next- if none of the angles are the same, how do I find the largest? I don't know if finding d helped at all so have I done the wrong thing?

 

[pka]

question: the sizes of the interior angles of a hexagon ABCDEF form an arithmetic sequence. the smallest interior angle of the hexagon is 75 degrees. find the size of the largest interior angle of the hexagon.

oh, okay! thanks, I'll try.
d = 18
sorry, I really don't know what to do next- if none of the angles are the same, how do I find the largest? I don't know if finding d helped at all so have

You are correct \(\displaystyle a=72~\&~d=18\)
We know that \(\displaystyle a_n=75+(n-1)(18)\) Isn"t the largest angle \(\displaystyle a_6~?\)

 

[bumblebee123]

a6 = 75 + ( 5 x 18 ) = 75 + 90 = 165 which is the correct answer

how did you know that the largest angle would be a6? Is it because 6 is the largest term out of all 6 terms?

75 + ( 6 - 1 ) ( 18 ) is going to make a bigger angle than 75 + ( 5 - 1) (18 )

 

[pka]

a6 = 75 + ( 5 x 18 ) = 75 + 90 = 165 which is the correct answer
how did you know that the largest angle would be a6? Is it because 6 is the largest term out of all 6 terms?
75 + ( 6 - 1 ) ( 18 ) is going to make a bigger angle than 75 + ( 5 - 1) (18 )

Because \(\displaystyle 18>0\) and \(\displaystyle a_n=75+18(n-1)\) then \(\displaystyle a_1<a_2<\cdots<a_6\).

 

[Dr.Peterson]

If you're not sure, all you need to do is to write out the six terms and see what you have. Those are the six angles.

You don't have to know all about a problem in order to work on it; you're just learning. That just means your work won't be as efficient as it would be if you had more experience. But working things out the slow way actually helps you to learn more about how things work.

 

[bumblebee123]

thanks, both of you have been really helpful!

 

[Denis]

Can anyone help to explain what to do? I'm really confused. Question:
the sizes of the interior angles of a hexagon ABCDEF form an arithmetic sequence.
the smallest interior angle of the hexagon is 75 degrees.
find the size of the largest interior angle of the hexagon.

Do you now "see" that the problem is another way of asking:
what is the nth term of an arithmetic series with a = 75, n = 6 and sum = 720 ?

 

[bumblebee123]

Do you now "see" that the problem is another way of asking:
what is the nth term of an arithmetic series with a = 75, n = 6 and sum = 720 ?

yes, that makes sense ( why couldn't they have just worded it like that?! haha )