Question

Triangle ABC is such that \[AB = 9\]cm, \[BC = 6\]cm, \[AC = 7.5\]cm. \[\vartriangle DEF\] is similar to \[\vartriangle ABC\]. If \[EF = 12\]cm then \[DE\]is:
A. 6
B. 16
C. 18
D. 15

Answer 1

Hint: Here we use the property of similar triangles which states that two similar triangles have ratio of their corresponding sides equal. We write the ratio of corresponding sides of two given similar triangles and substitute the values of lengths to obtain the length of side DE.
* Two triangles are said to be similar if the triangles have congruent corresponding angles.
* Ratio of any two numbers a and b can be written as\[a:b\]. We can write the ratio in the form of fraction as \[\dfrac{a}{b}\].

Complete step-by-step answer:
We have \[\vartriangle DEF\] similar to\[\vartriangle ABC\].
We draw diagrams for two similar triangles.

Corresponding sides of \[\vartriangle ABC\]and \[\vartriangle DEF\]are:
AB is corresponding to DE, BC is corresponding to EF and AC is corresponding to DF.
Since we know the property of similar triangles states that the ratio of corresponding sides of similar triangles is equal.
We write the ratio of corresponding sides equal
\[\dfrac{{AB}}{{BC}} = \dfrac{{DE}}{{EF}}\]
Now we know that \[AB = 9\]cm, \[BC = 6\]cm and \[EF = 12\]cm
Substitute the values in the ratio.
\[ \Rightarrow \dfrac{9}{6} = \dfrac{{DE}}{{12}}\]
Cross multiply the denominator of RHS to numerator of LHS
\[ \Rightarrow \dfrac{{9 \times 12}}{6} = DE\]
\[ \Rightarrow \dfrac{{9 \times 6 \times 2}}{6} = DE\]
Cancel out the same terms from numerator and denominator.
\[ \Rightarrow 18 = DE\]
Therefore, the length of the side DE is 18cm.

So, option C is correct.

Note: Students might make mistakes while writing the corresponding sides of the similar triangles, you can always take help of the name of the triangles that are given similar. We move from left to right and make pairs of the sides, example \[\vartriangle ABC \sim \vartriangle DEF\], we write side AB corresponds to DE, BC corresponds to EF and CA corresponds to FD.