In a triangle with angles measuring a, b, and c degrees, the mean of b and c is a....

[helping]

This is one of the questions given on an optional worksheet, that I would like to explain to my daughter.

In a triangle with angles measuring a, b, and c degrees, the mean of b and c is a. What is the value of a.

Since mean is average and to find average of n numbers we add the numbers and then divide by n, is the solution to above problem: a = (b+c)/2?


Thanks

 

[Mrspi]

This is one of the questions given on an optional worksheet, that I would like to explain to my daughter.

In a triangle with angles measuring a, b, and c degrees, the mean of b and c is a. What is the value of a.

Since mean is average and to find average of n numbers we add the numbers and then divide by n, is the solution to above problem: a = (b+c)/2?


Thanks

Ok...suppose you let the measures of the three angles of the triangle be a, b, and c.

What do you know about the SUM of the three angles of any triangle?

Can you write an equation based on that fact?

You are also given THIS fact: the mean of b and c is a. How do you find the mean of two numbers? Can you write ANOTHER equation based on this fact?

Now, you should have two equations which will allow you to easily solve for the value of "a".

Please show us what you've tried. This is not really a very difficult problem, and we'd be happy to be more helpful if we see what you have attempted.

 

[wjm11]

This is one of the questions given on an optional worksheet, that I would like to explain to my daughter.

In a triangle with angles measuring a, b, and c degrees, the mean of b and c is a. What is the value of a.

Since mean is average and to find average of n numbers we add the numbers and then divide by n, is the solution to above problem: a = (b+c)/2?


Thanks

You are off to a good start. The other thing you need to remember is that the sum of the three angles must equal 180 degrees. So a + b + c = 180. If we subtract "a" from each side of the equation, we get b + c = 180 - a. We now have two equations:

a = (b+c)/2
(b + c) = (180 - a)

Can you proceed from here?
(Oops! Mrs. Pi beat me to it...)

 

[helping]

You are off to a good start. The other thing you need to remember is that the sum of the three angles must equal 180 degrees. So a + b + c = 180. If we subtract "a" from each side of the equation, we get b + c = 180 - a. We now have two equations:

a = (b+c)/2
(b + c) = (180 - a)

Can you proceed from here?
(Oops! Mrs. Pi beat me to it...)

I understand that the total of all angles in a triangle = 180 degrees.

What I don't understand is why is there a need for the 2nd equation.

Why writing a = (b+c)/2 is not sufficient?

Not sure how to proceed...

 

[wjm11]

I understand that the total of all angles in a triangle = 180 degrees.

What I don't understand is why is there a need for the 2nd equation.

Why writing a = (b+c)/2 is not sufficient?

Not sure how to proceed...

In this problem, "a" has a unique value. You have not found/stated that value. You need to combine the two equations (to eliminate the variables b and c) and solve for a.

 

[helping]

In this problem, "a" has a unique value. You have not found/stated that value. You need to combine the two equations (to eliminate the variables b and c) and solve for a.

Being that this problem was from the optional worksheet she was not taught how to combine the two fractions yet, so it would be helpful if you can show us how to do that.

Thanks

 

[wjm11]

Being that this problem was from the optional worksheet she was not taught how to combine the two fractions yet, so it would be helpful if you can show us how to do that.

Thanks

So a + b + c = 180. If we subtract "a" from each side of the equation, we get b + c = 180 - a. We now have two equations:

a = (b+c)/2
(b + c) = (180 - a)

Next, notice that we have the expression (b + c) in both equations. That means we can substitute the value of (b + c) from the second equation into the first equation:

a = (180 - a)/2
a = 90 - a/2

Can you proceed now?

 

[helping]

I think so...

a = 90 - a/2

a/1 + a/2 = 90
3a/2=90
3a = 180
a=60

but I have a few questions...

So a + b + c = 180. If we subtract "a" from each side of the equation, we get b + c = 180 - a. We now have two equations:

What is the purpose of changing a + b + c = 180 to b + c = 180 - a?

a = (b+c)/2
(b + c) = (180 - a)

Next, notice that we have the expression (b + c) in both equations. That means we can substitute the value of (b + c) from the second equation into the first equation:

You mean not substitute but eliminate these two expressions right? If so, is this a rule that I can use in the future - whenever there are two identical expressions in two equations they can be eliminated?

Thanks

 

[wjm11]

I think so...

a = 90 - a/2

a/1 + a/2 = 90
3a/2=90
3a = 180
a=60

but I have a few questions...



What is the purpose of changing a + b + c = 180 to b + c = 180 - a?



You mean not substitute but eliminate these two expressions right? If so, is this a rule that I can use in the future - whenever there are two identical expressions in two equations they can be eliminated?

Thanks

First, the purpose of subtracting "a" was to get (b + c) by itself. We then knew that (b + c) was equal to (180 - a), right? Knowing that, we were able to substitute (180 - a) in place of (b + c) in the first equation. It was a SUBSTITUTION. We did not eliminate anything. We made (b+c) disappear by substituting an expression of equal value in its place.

 

[mmm4444bot]

whenever there are two identical expressions in two equations they can be eliminated?

If each of those two expressions can be isolated as terms in their respective equations, then the answer is yes. They can be eliminated by subtracting one equation from the other equation, leading to a new, third equation in which the identical expression no longer appears.

But, the method discussed above is the Substitution Method, not the Elimination Method.

Both methods lead to the same equation: 3a = 180

Maybe your daughter was taught both methods, but forgot them.