Please hELP.

Question

The sum of the angles in a triangle is 180 degrees. The middle size angle is 12 less than three times the smallest angle. The largest angle is twice the sum of the smaller two angles. Define your variable: Let x = ____ (Write expressions for the 3 unknown in terms of x [simplify if possible])

Answer 1

Hi Jessica,

 

Problem sounds like a mouthful !!

 

We'll take it step-by-step and all will be clear:

    1.  Let's call our smallest angle (x).

    2.  Let's call our middle angle (y).

    3.  We can express y in terms of x:

             3x -12 [12 less than 3 times x].

    4.  Let's call our largest angle (z).

    5.  We can express z in terms of x and y:

             2(x + y) [twice the sum of x and y).

    6.  We can express z in terms of x by replacing y

         with our x expression for y in our z expression:

             2(x + y) = 2(x + 3x - 12)

             2(x + 3x -12) = 2(4x - 12) = 8x -24

    7.  Now we have all three unknowns expressed in

         terms of just one unknown:

             x; y = 3x - 12; z  = 8x - 24

    8.  We're now ready to plug in all our x expressions

         into the triangle sum formula and solve for x

         and then using value for x,  we can determine

         values for y and z:

             x + 3x - 12 + 8x - 24 = 180

             12x - 36 = 180

             12x = 180 + 36

             12x = 216

                x = 216 / 12

                x = 18 degrees (smallest angle)

                y = 3x - 12

                y = 3(18) - 12

                y = 54 - 12

                y = 42 degrees (middle angle)

                z = 8x - 24

                z = 8(18) - 24

                z = 144 - 24

                z = 120 degrees (largest angle)

    9.  We can verify our three angle answers by

         seeing if they all add up 180 degrees:

             18 + 42 + 120 = 180

             60 + 120 = 180

             180 = 180 (angles add up to 180 degrees)

 

The trick to solving this problem was realizing that we could express all three unknowns in terms of just one unknown based upon the given information.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.

Answer 2

s represents measure of smallest angle

3s - 12 represents the measure of the middle angle

2(s + 3s - 12) represents the measure of the largest angle

 

s + 3s - 12 + 2(s + 3s - 12) = 180          distribute

s + 3s - 12 + 2s + 6s - 24 = 180             combine like terms

12s - 36 = 180                                        add 36 to both sides

12s = 216                                               divide both sides by 12

s = 18

 

Check:

18 + 3(18) - 12 + 2(18 + 3(18) - 12)

18 + 54 - 12 + 2(18 + 54 - 12)

18 + 54 - 12 + 2(60)

18 + 54 - 12 + 120

180