# \[\Delta ABC\sim \Delta XYZ\]. The ratio of corresponding sides of the triangles AB : XY = 2 : 3 . If BC = 5 cm. Find the value of YZ.

Last updated date: 23rd Mar 2023

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Answer

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Hint: To solve the question, we have to apply the property of similar triangles to calculate the value of YZ.

Complete step-by-step Solution:

\[\Delta ABC\sim \Delta XYZ\] symbolises that the triangles ABC and XYZ are similar triangles, which implies that the ratio of all the corresponding side of the given triangles is equal.

AB, BC, CA of triangle ABC are corresponding sides of XY, YZ, ZX of triangle XYZ respectively.

\[\Rightarrow \dfrac{AB}{XY}=\dfrac{BC}{YZ}=\dfrac{CA}{ZX}\]

The given value of side BC of triangle ABC is equal to 5 cm.

The given ratio of side AB of triangle ABC to side XY of triangle XYZ is equal to 2 : 3

By substituting the given values in the above expression, we get

\[\dfrac{2}{3}=\dfrac{5}{YZ}=\dfrac{CA}{ZX}\]

By solving the first part of the expression \[\dfrac{2}{3}=\dfrac{5}{YZ}\] we get,

\[2YZ=5\times 3\]

\[YZ=\dfrac{15}{2}=7.5\] cm.

Thus, the value of YZ is equal to 7.5 cm.

Note: The possibility of mistake can be not applying the similar triangles property which is required to arrive at the solution. The other possibility of mistake can be misinterpreting the symbol of similarity to the symbol of congruence. The symbol for a similar triangle is one negation sign and the symbol for congruent triangle is two negation signs. The alternative method of solving the question can be by applying the direct formula for calculating \[YZ=\dfrac{XY}{AB}\times BC\]. Thus, the answer can be calculated quickly.

Complete step-by-step Solution:

\[\Delta ABC\sim \Delta XYZ\] symbolises that the triangles ABC and XYZ are similar triangles, which implies that the ratio of all the corresponding side of the given triangles is equal.

AB, BC, CA of triangle ABC are corresponding sides of XY, YZ, ZX of triangle XYZ respectively.

\[\Rightarrow \dfrac{AB}{XY}=\dfrac{BC}{YZ}=\dfrac{CA}{ZX}\]

The given value of side BC of triangle ABC is equal to 5 cm.

The given ratio of side AB of triangle ABC to side XY of triangle XYZ is equal to 2 : 3

By substituting the given values in the above expression, we get

\[\dfrac{2}{3}=\dfrac{5}{YZ}=\dfrac{CA}{ZX}\]

By solving the first part of the expression \[\dfrac{2}{3}=\dfrac{5}{YZ}\] we get,

\[2YZ=5\times 3\]

\[YZ=\dfrac{15}{2}=7.5\] cm.

Thus, the value of YZ is equal to 7.5 cm.

Note: The possibility of mistake can be not applying the similar triangles property which is required to arrive at the solution. The other possibility of mistake can be misinterpreting the symbol of similarity to the symbol of congruence. The symbol for a similar triangle is one negation sign and the symbol for congruent triangle is two negation signs. The alternative method of solving the question can be by applying the direct formula for calculating \[YZ=\dfrac{XY}{AB}\times BC\]. Thus, the answer can be calculated quickly.

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