Answer
456k+ views
Hint: The formula for selecting r different things out of n different things is given by \[{}^{n}{{C}_{r}}\] .
The formula for evaluating probability of any event is
P \[=\dfrac{Favorable\ outcomes}{Total\ outcomes}\] .
Equilateral triangle is a type of triangle in which all three sides are of equal length.
Complete Step-by-step answer:
As mentioned in the question, we need to know how many equilateral triangles are there if we join 3 vertices of a regular hexagon.
Another fact about equilateral triangles is that all the angles of an equilateral triangle is equal to \[{{60}^{\circ }}\] .
Using this fact, we can calculate that there can be only two triangles that are equilateral when we join three vertices of a regular hexagon.
In the figure, the two possible equilateral triangles are marked.
Hence, the total outcomes for the event is given by
\[{}^{6}{{C}_{3}}\] (As the total vertices in a regular hexagon are 6 and the number of vertices that we need to pick are 3)
\[\begin{align}
& {}^{6}{{C}_{3}}=\dfrac{6!}{3!\left( 6-3 \right)!} \\
& =20 \\
\end{align}\]
Hence, the total outcomes are 20.
Therefore, the probability of the event to take place is
\[\begin{align}
& P=\dfrac{Favorable\ outcomes}{Total\ outcomes} \\
& P=\dfrac{2}{20} \\
\end{align}\]
Hence, the probability is \[\dfrac{1}{10}\] .
Note: The students can make an error in evaluating the number of equilateral triangles that are formed when three vertices of the regular hexagon are selected at random.
Without knowing the property of equilateral triangles that equilateral triangles are those with all the angles of an equilateral triangle is equal to \[{{60}^{\circ }}\] , they would not get to the right answer.
The formula for evaluating probability of any event is
P \[=\dfrac{Favorable\ outcomes}{Total\ outcomes}\] .
Equilateral triangle is a type of triangle in which all three sides are of equal length.
Complete Step-by-step answer:
As mentioned in the question, we need to know how many equilateral triangles are there if we join 3 vertices of a regular hexagon.
Another fact about equilateral triangles is that all the angles of an equilateral triangle is equal to \[{{60}^{\circ }}\] .
Using this fact, we can calculate that there can be only two triangles that are equilateral when we join three vertices of a regular hexagon.
![seo images](https://www.vedantu.com/question-sets/95933cd9-89ac-497b-9468-bca89ce5f9aa8809076919295688125.png)
![seo images](https://www.vedantu.com/question-sets/72f6a564-b9e4-48ae-9c28-05676a28a7118195243350284044329.png)
In the figure, the two possible equilateral triangles are marked.
Hence, the total outcomes for the event is given by
\[{}^{6}{{C}_{3}}\] (As the total vertices in a regular hexagon are 6 and the number of vertices that we need to pick are 3)
\[\begin{align}
& {}^{6}{{C}_{3}}=\dfrac{6!}{3!\left( 6-3 \right)!} \\
& =20 \\
\end{align}\]
Hence, the total outcomes are 20.
Therefore, the probability of the event to take place is
\[\begin{align}
& P=\dfrac{Favorable\ outcomes}{Total\ outcomes} \\
& P=\dfrac{2}{20} \\
\end{align}\]
Hence, the probability is \[\dfrac{1}{10}\] .
Note: The students can make an error in evaluating the number of equilateral triangles that are formed when three vertices of the regular hexagon are selected at random.
Without knowing the property of equilateral triangles that equilateral triangles are those with all the angles of an equilateral triangle is equal to \[{{60}^{\circ }}\] , they would not get to the right answer.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)