The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. $\frac{\mathrm{PQ}}{\mathrm{AB}}$ is equal to
Explanation
Given that the perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively, we can write the ratio of their perimeters as $\\frac{48}{56} = \\frac{6}{7}$. Since the triangles are similar, the ratio of their corresponding sides is equal to the ratio of their perimeters. Let $x$ be the length of side AB. Then, the length of side PQ is $\\frac{6}{7}x$. The perimeter of triangle ABC is $x + x + x = 3x$, and the perimeter of triangle PQR is $\\frac{6}{7}x + \\frac{6}{7}x + \\frac{6}{7}x = \\frac{18}{7}x$. We are given that the perimeter of triangle ABC is 56 cm, so we can write $3x = 56$. Solving for $x$, we get $x = \\frac{56}{3}$. Now, we can find the length of side PQ by substituting the value of $x$ into the expression $\\frac{6}{7}x$. We get $\\frac{6}{7} \\times \\frac{56}{3} = \\frac{336}{21} = 16$. Therefore, the ratio of the length of side PQ to the length of side AB is $\\frac{16}{56/3} = \\frac{16}{56} \\times \\frac{3}{1} = \\frac{48}{56} = \\frac{6}{7}$
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