If R is the midpoint of QS, RS = 2x - 4, ST = 4x - 1, and RT = 8x + 43, find QS.

Understand the Problem

The question is asking to find the value of segment QS given that R is the midpoint of QS and provides expressions for segments RS, ST, and RT. To solve it, we will set up equations based on the relationships between the segments.

Answer

The value of segment $ QS $ is $ 68 $.

Steps to Solve

Identify Known Relationships

Since ( R ) is the midpoint of ( QS ), we know that: $$ QR = RS $$

Set Up Equations for Each Segment

From the problem, we have:

( RS = 2x - 4 )
( ST = 4x - 1 )
( RT = 8x - 43 )

Also, since ( R ) is the midpoint: $$ QR = RS $$ Thus, we can express ( QR ) as: $$ QR = \frac{QS}{2} $$

Express ( QS ) in Terms of ( x )

Since ( QS ) can be expressed as: $$ QS = QR + RS = RS + RS = 2RS $$ So: $$ QS = 2(2x - 4) = 4x - 8 $$

Determine ( ST ) in terms of ( x )

We set ( RT ) as the sum of ( RS ) and ( ST ): $$ RT = RS + ST $$ Substituting the respective expressions: $$ 8x - 43 = (2x - 4) + (4x - 1) $$

Solve for ( x )

Combine like terms: $$ 8x - 43 = 2x - 4 + 4x - 1 $$ $$ 8x - 43 = 6x - 5 $$

Rearranging gives: $$ 8x - 6x = -5 + 43 $$

This simplifies to: $$ 2x = 38 $$ Thus: $$ x = 19 $$

Calculate ( QS )

Substituting ( x = 19 ) back into the expression for ( QS ): $$ QS = 4x - 8 = 4(19) - 8 $$ $$ QS = 76 - 8 = 68 $$

The value of segment ( QS ) is ( 68 ).

More Information

This problem involved using relationships between segments, particularly the concept of midpoints. By expressing one segment in terms of others and solving for the variable ( x ), we were able to find the total length of segment ( QS ).

Tips

Forgetting that ( R ) is the midpoint can lead to incorrect equations.
Not combining like terms correctly when simplifying.

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