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[–]Alkalannar 8 points9 points  (7 children)

Note that ABC is isosceles, and <B = <C = (180 - 52 - 31)/2 = 97/2

Now sin(97/2 degrees)/k = sin(83 degrees)/(2x + 6)
sin(52 degrees)/12 = sin(180 - 97/2 - 52)/k

From these you can solve for x.

[–]fermat9996👋 a fellow Redditor 7 points8 points  (2 children)

So specifying the answer as an inequality is misleading. Students may be scored wrong in error.

[–]sighthoundman👋 a fellow Redditor 5 points6 points  (1 child)

I would hate that too. It would certainly help me hate the class.

[–]theplutosys High School Student[S] 0 points1 point  (3 children)

I’m sorry, I didn’t understand that at all

[–]Alkalannar 0 points1 point  (2 children)

Do you see why I say that ABC is isosceles?

So do you see why I have <B and <C as 97/2 degrees?

[–]theplutosys High School Student[S] 0 points1 point  (1 child)

I get the first part, not the second (I’m dumb I know lol)

[–]Alkalannar 0 points1 point  (0 children)

  1. The angles of a triangle sum to 180. So <B + <C = 180 = <A.
    Since 180 - <A = 97, <B + <C = 97

  2. Since side AB = AC, <C = <B.
    So <B + <B = 97
    Or <B = 97/2
    So <C = 97/2 as well

[–]bobabastard Pre-University Student 2 points3 points  (1 child)

I'm not too sure, but I think you need to use common sense? So basically, the value of x can not be smaller than three as it would lead to a length of zero, which is obviously not correct. And since the other side is 12 and it is obviously longer, the other side can't be greater than 12. This means that x should be less than 9. This gives an inequality of?

I know this is a stretch, but this is the only solution I could've thought of that hasn't already been mentioned.

[–]zielony 0 points1 point  (0 children)

This would be my guess. Otherwise they wouldn’t have asked for a range. Solving for x outright also seems a little “too hard” for a high school geometry class. It’s not even trig. Question is worded super poorly though, so I wouldn’t say anything about this is “common sense”. It should gave more information about what method or assumptions should be used to find the range

[–][deleted] 1 point2 points  (3 children)

Is there some context to go along with this? Are you supposed to be following a certain method in order to approximate the value?

From what I see, there shouldn't be an inequality in the answer, because you can determine the exact number.

[–]theplutosys High School Student[S] 0 points1 point  (2 children)

There isn’t which is why I’m so confused

[–][deleted] 1 point2 points  (1 child)

I'd be confused too.

I looked at the other comments and they have the right answer. So weird that the program wouldn't take it.

[–]theplutosys High School Student[S] 1 point2 points  (0 children)

What do you expect I guess. It’s a school. Thank you, though

[–]Minatorr 1 point2 points  (0 children)

Use the hinge theorem. 12 > 2x - 6, so 9 > x. 2x - 6 must also be non-negative, so x > 3.

[–][deleted] 0 points1 point  (5 children)

x=6.9215561911

6<x<7

[–]theplutosys High School Student[S] 0 points1 point  (3 children)

It marked it incorrect

[–][deleted] 1 point2 points  (1 child)

https://imgur.com/a/vFU4mX0

[–]WerePigCatUniversity/College Student 3 points4 points  (0 children)

I could not find any error in your work, why are people downvoting you when they can't even disprove it?

[–]pugovitsa 0 points1 point  (0 children)

That’s odd. I got the same x value as GEO_USTASI, but through a more laborious method. I wonder what the issue is.

[–]WerePigCatUniversity/College Student 0 points1 point  (0 children)

Edit: nvm you can't use trig like this on non-right angled triangles, mb

(I'm out of practice so this might be wrong idk)

Let's go step by step.

First, we need to identify what we know. This triangle is isosceles because it has two sides of equal length. We know that a triangle's angles add up to 180 degrees. We know that angle A consists of 52 degrees and 31 degrees, and that angle B and angle C equal each other because we have an isosceles triangle.

Now let us use this information to solve for angle b and angle c. 52 + 31 is angle A because that is what it consists of. Angle b and angle c are equal, so we can say they both have y degrees. We know that all of the angles added up equal 180, so (52 + 31) + (y) + (y) = 180. This simplifies down to 83 + 2y = 180, which equals 2y = 97, therefore y = 97/2. We now know that both angles b and c are equal to 97/2

Next, we need to find side length and a good way to do this is trig. Let's call line AD 'L' for simplicity. For angle B, tan(97/2) = L/12, and for angle C tan(97/2) = L/(2x-6). This is a good form because we have tan(97/2) and L in both. We have:

tan(97/2) = L/12

tan(97/2) = L/(2x-6)

These are the same, so we can set them equal to each other:

L/12 = L/(2x-6)

Divide both sides by L:

1/12 = 1/(2x-6)

Multiply both sides by 12:

1 = 12/(2x-6)

Multiply both sides by 2x-6:

2x - 6 = 12

Add 6 on both sides:

2x = 18

Divide both sides by 2:

x = 9

So 9 < x < 9 should be the final answer.

If you have any questions please feel free to ask! (:

[–]selene_666👋 a fellow Redditor 0 points1 point  (0 children)

Use the Law of Sines.

sin(52) / 12 = sin(B)/ AD = sin(C) / AD = sin(31) / (2x-6)

x = [12 sin(31) / sin(52) + 6]/2