Answer
Verified
430.5k+ views
Hint: For the solution of this question, we will be using the Pythagoras Theorem.
Pythagoras Theorem states that,
\[\Rightarrow {{(Hypotenuse)}^{2}}={{(Perpendicular)}^{2}}+{{(Base)}^{2}}\]
Now, since the triangle is an equilateral triangle, its altitude will bisect its base, and also, it will be perpendicular to its base.
So, as we get us Hypotenuse, Base and Perpendicular, we will put the values of Hypotenuse, Base and Perpendicular in the theorem to find the length of the altitude of the equilateral triangle.
The altitude will act like the perpendicular of the triangle which is equilateral.
Complete step by step answer:
Let the equilateral triangle be ABC.
We will now draw the altitude AD,
Now, in right angled ∆ABD, we will use the Pythagoras theorem as follows,
\[\begin{align}
& \Rightarrow {{(H)}^{2}}={{(P)}^{2}}+{{(B)}^{2}} \\
& \Rightarrow {{(AB)}^{2}}={{(AD)}^{2}}+{{(BD)}^{2}} \\
& \Rightarrow {{(2a)}^{2}}={{(AD)}^{2}}+{{(a)}^{2}} \\
& \Rightarrow {{(2a)}^{2}}-{{(a)}^{2}}={{(AD)}^{2}} \\
& \Rightarrow 4{{a}^{2}}-{{a}^{2}}={{(AD)}^{2}} \\
& \Rightarrow 3{{a}^{2}}={{(AD)}^{2}} \\
& \Rightarrow \sqrt{(3{{a}^{2}})}=(AD) \\
& \Rightarrow \sqrt{3}a=AD \\
\end{align}\].
So, the measure of AD is \[\sqrt{3}a\] .
So, the correct answer is “Option B”.
Note: Let us now understand about Altitude and Median.
ALTITUDE: Altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side. It is also known as the height of the triangle.
MEDIAN: A median of triangle is a line segment that joins a vertex to the mid-point of the side that is opposite to that vertex.
In the figure given below, in ∆ABC, AD is the altitude of the triangle ABC and AE is the median that divides BC into two equal halves, that is, BE = EC.
Pythagoras Theorem states that,
\[\Rightarrow {{(Hypotenuse)}^{2}}={{(Perpendicular)}^{2}}+{{(Base)}^{2}}\]
Now, since the triangle is an equilateral triangle, its altitude will bisect its base, and also, it will be perpendicular to its base.
So, as we get us Hypotenuse, Base and Perpendicular, we will put the values of Hypotenuse, Base and Perpendicular in the theorem to find the length of the altitude of the equilateral triangle.
The altitude will act like the perpendicular of the triangle which is equilateral.
Complete step by step answer:
Let the equilateral triangle be ABC.
We will now draw the altitude AD,
Now, in right angled ∆ABD, we will use the Pythagoras theorem as follows,
\[\begin{align}
& \Rightarrow {{(H)}^{2}}={{(P)}^{2}}+{{(B)}^{2}} \\
& \Rightarrow {{(AB)}^{2}}={{(AD)}^{2}}+{{(BD)}^{2}} \\
& \Rightarrow {{(2a)}^{2}}={{(AD)}^{2}}+{{(a)}^{2}} \\
& \Rightarrow {{(2a)}^{2}}-{{(a)}^{2}}={{(AD)}^{2}} \\
& \Rightarrow 4{{a}^{2}}-{{a}^{2}}={{(AD)}^{2}} \\
& \Rightarrow 3{{a}^{2}}={{(AD)}^{2}} \\
& \Rightarrow \sqrt{(3{{a}^{2}})}=(AD) \\
& \Rightarrow \sqrt{3}a=AD \\
\end{align}\].
So, the measure of AD is \[\sqrt{3}a\] .
So, the correct answer is “Option B”.
Note: Let us now understand about Altitude and Median.
ALTITUDE: Altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side. It is also known as the height of the triangle.
MEDIAN: A median of triangle is a line segment that joins a vertex to the mid-point of the side that is opposite to that vertex.
In the figure given below, in ∆ABC, AD is the altitude of the triangle ABC and AE is the median that divides BC into two equal halves, that is, BE = EC.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE