Question 1034432
The problem says that triangles {{{MNP}}} and {{{XYZ}}} are similar.
The corresponding vertices are given in the same order,
so {{{M}}} correspond to {{{X}}}} ,
and the angles at those vertices are congruent (have the same measure).
The same is true for {{{N}}} and {{{Y}} , and for {{{P}}} and {{{Z}}} .
Sides {{{MP}}} and {{{XZ}}} are corresponding sides.
So are sides {{{NP}}} and {{{YZ}}} .
Besides angles having the same measure,
the fact that the triangles are similar means that
triangle {{{MNP}}} is a scaled up or scaled down version of triangle {{{XYZ}}} .
The scaling factor ismthe ratio of lengths of the corresponding sides:
{{{MP/XZ=NP/YZ}}} .
On the figure, the lengths of those sides are shown as:
{{{MP=x+5}}}
{{{NP=4x-10}}}
{{{XZ=30}}}
{{{YZ=75}}} .
So, {{{(x+5)/30=(4x-10)/75}}}
From that equation, we solve for {{{x}}} .
First, we eliminate denominators by multiplying times a common multiple of the denominators,
{{{2*75=150=5*30}}} .
{{{(x+5)/30=(4x-10)/75}}}<-->{{{150*(x+5)/30=150*(4x-10)/75}}}<-->{{{5*(x+5)=2*(4x-10)}}}<-->{{{5x+25=8x-20)}}} .
The rest is easy:
{{{5x+25=8x-20)}}}<-->{{{5x+25-5x+20=8x-20-5x+20)}}}<-->{{{25+20=8x-5x)}}}<-->{{{45=3x)}}}<-->{{{45/3=x}}},-->{{{highlight(x=15)}}}