# Is it possible to form a triangle with the given side lengths? 3,4,6

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In order to determine if a triangle can be formed with 3 sides we must use the Triangle Inequality Theorem.

*Triangle Inequality Theorem states that any side of a triangle must be shorter than the sum of the other two sides.*

Let the sides of the triangle be:

a = 3; b = 4; c = 6

This means `b+cgta` `rArr 4+6 > 3`

`a+b > crArr 3+4>6`

`a+c > brArr 3+6 > 4`

Since these are all true, the lengths of 3,4, and 6 will form a triangle.

**Sources:**

Given a triangle with sides a=3, b=4 and c=6 and angles A opposite to side a, B opposite to side b and C opposite to side c, apply the law of cosines to determine angle A.

`cosA=(b^2+c^2-a^2)/(2bc)`

Substitute 4 for b, 6 for c and 3 for a.

`cosA=(4^2+6^2-3^2)/((2)(4)(6))=(16+36-9)/48=43/48=0.8958`

`A=cos^-1(0.8958)=26.4`o

Apply the law of sines to determine angle B

`sinA/a=sinB/b`

Substitute 0.44 for sin A, 3 for a and 4 for b and solve for `sinB`

`sinB=((0.44)(4))/3=0.5867`

`B=sin^-1(0.5867)=35.9`o

Find angle C using: `A+B+C=180`o

`C=180-35.9-26.4=117.7`o

**Thus it is possible to form a triangle with sides 3, 4 and 6.** The angles are A=26.4 degrees, B=36.9 degrees and C=117.7 degrees.

In order to decided wheter the sides given can truly form a triangle, you must use the **Triangle Inequality Theorem**

You label the 3 sides: side a, side b, side c. It does not matter on which side a, b, or c.

`Delta ` A+B>C

B+C>A

A+C>B

For this example:

We could label this triangle with sides

side a= 3

side b= 4

side c =6

To determine if it is a triangle, you have to see if **all the inequalites are true**

3+4>6 Correct

4+6>3

Correct3+6>3

Correct

**All sides do fit the Triange Inequality Theorem, therefore, it is possible to form a triangle with these side lengths.**

*If you have any other questions on this topic. Feel free to contact me. Hope I have helped you.*