#### Xuequinox wrote:

(5.2.1.) #67:I'm going to explain this in detail, forgive me if this is annoying.

"What are the possible side lengths for sidein the triangle /(at the right)/? Show how you know."ML

Given information:MS= 10",SL= 4". Anglecan be observed to have anMSLobtuse angle.

Edit: All I've been able to assume is that ML could be 12.

a^2 + b^2 = c^2

where

(a, b and c are all greater than 0 because they are side lengths)where

*a*,*b*, and*c*are sides of a triangleThat's the Pythagorean Theorem, which is the core for solving problems like these.

For example, a triangle with sides of 3, 4, and 5 can be considered a right triangle:

3^2+4^2=5^2

9+16=25

25=25

If we take this one step further, we can find out that:9+16=25

25=25

**1.**if a^2 + b^2 < c^2, then the triangle is obtuse. generally you have to use the longest side length for c to make this work.

**2.**if a^2 + b^2 > c^2, then the triangle is acute.

Great, i'm going off-topic.

Let's look at the given first:

__MS__= 10"__SL__= 4"- Angle
can be observed to have an__MSL__*obtuse angle*

Because MSL has an obtuse angle, (MS)^2+(SL)^2 < (ML)^2.

So let's plug in the values for MS and SL:

10^2 + 4^2 < ML^2

100 + 16 < ML^2

116 < ML^2

(taking the square root of both sides)

which approximates to 10.7703296

100 + 16 < ML^2

116 < ML^2

(taking the square root of both sides)

**sqrt(116) < ML**which approximates to 10.7703296

That's only half of it though.

We need to find a

*maximum*value for ML as well, because think about it:

we can put MS and SL into a single line, can we not? Since a straight line cannot form a triangle anyway, we can take the addition of the two known sides to find the maximum. (sorry, idek if this part will make sense)

So to find our maximum value of ML, we simply add the other two side lengths we know.

ML < MS + SL

ML < 10 + 4

ML < 10 + 4

__ML < 14__So if we put these two together, our final answer will be:

**sqrt(116) < ML < 14**

p.s.: i forgot to mention that there's a thing called the Triangle Inequality Theorem.

It's important to know if you want to solve problems like this in the future.

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