Question 1193077


Given: 

{{{RM = RN = 3x + 1}}}
{{{ST = 7x - 2}}}
{{{m}}} < {{{R = 60}}}°

Find: {{{x}}}, {{{RM}}}, and {{{ST}}}

if you draw the triangle and label the sides you will see that the smaller triangle{{{ RMN}}} is an isosceles triangle because two of its sides are equal. So the base angles have to be {{{congruent}}}.

we can find the missing angles because we have measure of angle {{{R = 60}}} and we know that sum of interior angles in a triangle is {{{180}}}.

Measure of angle {{{RMN}}}= measure of angle {{{RNM= (180-60)/2=60}}}

you will see that all the three angles in triangle {{{RMN}}} are {{{60}}} each. 

So it is an {{{equilateral}}} triangle. 

Then the 3rd side {{{MN}}} is equal to the first two sides{{{ RM}}} and {{{RN}}}.

so we now have {{{MN= 4x +1}}}

using triangle mid segment theorem, we know that the midsegment {{{MN}}} ( line connecting the midpoints of two sides of a triangle is midsegment) is parallel to {{{ST}}} and it’s length is half of the length of {{{ST}}}

{{{MN =  ST/2}}} or

{{{ST = 2 MN}}}}

plug in the values

{{{7x -2 = 2(3x + 1)}}}

{{{7x -2 = 6x + 6}}}

{{{7x -6x = 2 + 6}}}

{{{x = 8}}}

then

{{{RM = 3*8 + 1=25}}}
{{{ST = 7*8 - 2=54}}}