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The perimeter of a triangle is the sum of its three sides. $△ABC$ has the sides $2x,$ $2x+4,$ and $2x+3.$ Adding these, we can create an algebraic expression that describes the perimeter of the triangle.
$P_{△ABC}=2x+(2x+4)+(2x+3)=6x+7 $
The triangle $△PQR$ has the sides $3x+1,$ $2x,$ and $2x.$ We can create an expression for the perimeter of this triangle as well.
$P_{△PQR}=(3x+1)+2x+2x=7x+1 $
Since the perimeters are equal, we can equate the expressions, and solve for $x.$
Knowing that $x=6,$ we can find the side lengths of both triangles by substituting this value into the corresponding expressions.

$6x+7=7x+1$

SubEqn$LHS−6x=RHS−6x$

$7=x+1$

RearrangeEqnRearrange equation

$x+1=7$

SubEqn$LHS−1=RHS−1$

$x=6$

Expression | $x=6$ | Side length |
---|---|---|

$2x$ | $2⋅6$ | $12$ |

$2x+3$ | $2⋅6+3$ | $15$ |

$2x+4$ | $2⋅6+4$ | $16$ |

$3x+1$ | $3⋅6+1$ | $19$ |

Therefore, $△ABC$'s side lengths are $12,$ $15$ and $16,$ and the side lengths of $△PQR$ are $12,$ $12$ and $19.$