7. find the value of x in the figure below if \\(\\overline{mp}\\) is parallel to \\(\\overline{ra}\\)…

Explanation:

Step1: Identify the theorem

Since ( \overline{MP} \parallel \overline{RA} ), by the Basic Proportionality Theorem (Thales' theorem), the line ( MP ) divides the sides ( AR ) and ( AA ) (wait, correction: the sides ( AR ) and ( AP )? Wait, looking at the figure, the triangle is ( ARA )? No, probably a typo, the vertices are ( R ), ( A ) (lower), and ( A ) (upper)? Wait, no, the figure has points ( R ), ( M ), ( A ) (upper) on the top side, and ( R ), ( A ) (lower), ( P ) on the bottom side? Wait, actually, the triangle is ( A ) (lower) - ( R ) - ( A ) (upper), with ( M ) on ( RA ) (upper) and ( P ) on ( A ) (lower) - ( A ) (upper) side? Wait, no, the correct interpretation is that ( MP ) is parallel to ( RA ), so the triangle has a line segment ( MP ) parallel to ( RA ), so by Thales' theorem, ( \frac{RM}{MA} = \frac{AP}{PA} )? Wait, no, let's look at the lengths: ( MA = 18 ), ( AP = 5 ), ( PA ) (wait, the side with length 20? Wait, the side from ( P ) to ( A ) (upper) is 20? Wait, no, the figure: ( R ) to ( M ) is ( x ), ( M ) to ( A ) (upper) is 18, ( P ) to ( A ) (upper) is 20, ( A ) (lower) to ( P ) is 5. So the triangle is ( A ) (lower) - ( R ) - ( A ) (upper), with ( M ) on ( RA ) (upper) and ( P ) on ( A ) (lower) - ( A ) (upper) side. So ( MP \parallel RA ), so by Thales' theorem, ( \frac{RM}{MA} = \frac{AP}{PA} )? Wait, no, the segments: ( AP = 5 ), ( PA ) (wait, the entire side from ( A ) (lower) to ( A ) (upper) is ( AP + PA = 5 + 20 = 25 )? Wait, no, the side from ( A ) (lower) to ( P ) is 5, and from ( P ) to ( A ) (upper) is 20, so total length ( AA = 5 + 20 = 25 ). Then ( MP ) is parallel to ( RA ), so in triangle ( ARA ) (wait, no, the triangle is ( A ) (lower) - ( R ) - ( A ) (upper), with ( M ) on ( RA ) (upper) and ( P ) on ( A ) (lower) - ( A ) (upper) side. So by Thales' theorem, ( \frac{RM}{MA} = \frac{AP}{PA} )? Wait, no, the correct ratio is ( \frac{RM}{MA} = \frac{AP}{PA} )? Wait, no, Thales' theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So the two sides are ( AR ) (from ( A ) (lower) to ( R )) and ( AA ) (from ( A ) (lower) to ( A ) (upper)). Wait, no, the sides are ( A ) (lower) - ( R ) and ( A ) (lower) - ( A ) (upper). Then ( MP ) is parallel to ( R ) - ( A ) (lower)? No, ( MP ) is parallel to ( RA ) (where ( RA ) is from ( R ) to ( A ) (lower)). So the triangle is ( A ) (lower) - ( R ) - ( A ) (upper), with ( M ) on ( R ) - ( A ) (upper) and ( P ) on ( A ) (lower) - ( A ) (upper). Then ( MP \parallel RA ) ( ( RA ) is from ( R ) to ( A ) (lower) ). So by Thales' theorem, ( \frac{RM}{MA} = \frac{AP}{PA} ). Wait, ( RM = x ), ( MA = 18 ), ( AP = 5 ), ( PA = 20 )? Wait, no, ( AP ) is 5, ( PA ) is 20? Wait, the side from ( A ) (lower) to ( P ) is 5, and from ( P ) to ( A ) (upper) is 20, so the ratio of ( AP ) to ( PA ) is ( 5:20 = 1:4 )? No, wait, the correct ratio is ( \frac{RM}{MA} = \frac{AP}{PA} )? Wait, no, Thales' theorem: if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So the two sides are ( A ) (lower) - ( R ) and ( A ) (lower) - ( A ) (upper). The line ( MP ) intersects ( A ) (lower) - ( R ) at ( M )? No, ( M ) is on ( R ) - ( A ) (upper), and ( P ) is on ( A ) (lower) - ( A ) (upper). So the triangle is ( A ) (lower) - ( R ) - ( A ) (upper), with ( M ) on ( R ) - ( A ) (upper) and ( P ) on ( A ) (lower) - ( A ) (upper). Then ( MP \parallel RA ) ( ( RA ) is from ( R ) to ( A ) (lower) ). So the segments: ( R ) - ( M ) is ( x ), ( M ) - ( A ) (upper) is 18, ( A ) (lower) - ( P ) is 5, ( P ) - ( A ) (upper) is 20. So by Thales' theorem, ( \frac{RM}{MA} = \frac{AP}{PA} )? Wait, ( AP = 5 ), ( PA = 20 ), so ( \frac{x}{18} = \frac{5}{20} )? No, that would be ( x = 18 \times \frac{5}{20} = 4.5 ). Wait, that makes sense. So the ratio is ( \frac{RM}{MA} = \frac{AP}{PA} ), where ( RM = x ), ( MA = 18 ), ( AP = 5 ), ( PA = 20 ). So ( \frac{x}{18} = \frac{5}{20} ).

Step2: Solve for ( x )

Simplify ( \frac{5}{20} = \frac{1}{4} ), so ( x = 18 \times \frac{1}{4} = 4.5 ).

Answer:

4.5 units