Question

Let R be the relation defined in the set A of all triangles as\[R=\left\{ \left( {{T}_{1}},{{T}_{2}} \right):{{T}_{1}}\,is\,similar\,to\,{{T}_{2}} \right\}\]. If R is an equivalence relation and there are three right angles triangles ${{T}_{1}}$ with sides 3,4,5; ${{T}_{2}}$ with sides 5,12,13 and ${{T}_{3}}$ sides 6,8,10. Then which of the following is true.
(a) ${{T}_{1}}$ is related to ${{T}_{2}}$.
(b) ${{T}_{2}}$ is related to ${{T}_{3}}$.
(c) ${{T}_{1}}$ is related to ${{T}_{3}}$.
(d) none of these.

Answer 1

Hint: In this question, we will use proportionality by the corresponding sides rule of similar triangles to check given options.

Complete step-by-step solution -
In a given question, we have set A of all triangles. Relation R is defined on A such as,\[R=\left\{ \left( {{T}_{1}},{{T}_{2}} \right):{{T}_{1}}\,is\,similar\,to\,{{T}_{2}} \right\}\]. We are given three right angled triangles ${{T}_{1}}$, ${{T}_{2}}$ and ${{T}_{3}}$ such that, ${{T}_{1}}$ have sides of units 3,4 and 5.
${{T}_{2}}$ have sides of units 6,8 and 10.
Now, we know that, when two triangles are similar, then the ratio of their corresponding sides are equal.
Here, in triangles ${{T}_{1}}$ and ${{T}_{2}}$, ratio of their corresponding sides are:
$\dfrac{3}{5},\dfrac{4}{12}\,and\,\dfrac{5}{13}$
On simplifying, we get, $\dfrac{3}{5},\dfrac{1}{2}\,and\,\dfrac{5}{13}$.
Clearly, ratios of corresponding sides of triangles ${{T}_{1}}$ and ${{T}_{2}}$ are not equal. Therefore, ${{T}_{1}}$ and ${{T}_{2}}$ are not similar triangles.
Therefore, \[\left( {{T}_{1}},{{T}_{2}} \right)\] does not belong to R.
That is ${{T}_{1}}$ is related to ${{T}_{2}}$.
Now, in triangles ${{T}_{2}}\,and\,{{T}_{3}}$, ratios of corresponding sides of ${{T}_{2}}\,and\,{{T}_{3}}$ are: $\dfrac{5}{6},\dfrac{12}{8}\,\,and\,\dfrac{13}{10}$.
Clearly ratios of corresponding sides of ${{T}_{2}}\,and\,{{T}_{3}}$ are not equal.
Therefore, ${{T}_{2}}\,and\,{{T}_{3}}$ are not similar triangles.
Therefore, $\left( {{T}_{2}}\,and\,{{T}_{3}} \right)$ does not belong to R.
That is, ${{T}_{2}}$ is not related to ${{T}_{3}}$.
Now, in triangle ${{T}_{1}}$ and ${{T}_{3}}$, ratios of corresponding sides of ${{T}_{1}}$ and ${{T}_{3}}$ are: $\dfrac{3}{6},\dfrac{4}{8}\,\,and\,\dfrac{5}{10}$.
On simplifying, we get, $\dfrac{1}{2},\dfrac{1}{2}\,\,and\,\dfrac{1}{2}$.
Clarify, ratios of corresponding sides of ${{T}_{1}}$ and ${{T}_{3}}$ are on similar triangles, by side-side-side rule.
Therefore, $\left( {{T}_{1}},{{T}_{3}} \right)$ belongs to R.
That is ${{T}_{1}}$ is related to ${{T}_{3}}$.
Hence, the correct answer is option (c).

Note: In this type of question, do not get confused that all ${{T}_{1}}$, ${{T}_{2}}$ and ${{T}_{3}}$ are right angle triangles so they will be similar. All right angles triangles are not similar.