an equilateral triangle of side 10cm
i)find distance from centroid to midpoint
ii)find distance from any vertex to centroid
Answers
Distance Between Centroid & vertex of equilateral triangle is \frac{x}{\sqrt{3}}
3
x
Step-by-step explanation:
Given: Centroid and vertex of Equilateral triangle
To find: Distance between centroid and vertex.
Centroid is the point of intersection of all 3 medians of triangle.
Also, In equilateral triangle median and Altitude are same.
A centroid divide a median in 2 : 1. ratio.
Let x be the length of equilateral triangle ABC,
CB = \frac{1}{2}\times AB=\frac{x}{2}
2
1
×AB=
2
x
BD is median as well as altitude.
Using Pythagoras theorem in ΔABD
AB² = AD² + BD²
x^2=(\frac{x}{2})^2+BD^2x
2
=(
2
x
)
2
+BD
2
BD^2=x^2-\frac{x^2}{4}BD
2
=x
2
−
4
x
2
BD^2=\frac{3x^2}{4}BD
2
=
4
3x
2
BD=\frac{\sqrt{3}x}{2}BD=
2
3
x
⇒ Length of median = \frac{\sqrt{3}x}{2}
2
3
x
Total part of ratio = 1 + 2 = 3
⇒ Distance between Centroid & vertex of equilateral triangle = \frac{2}{3}\times\frac{\sqrt{3}x}{2}\:=\:\frac{x}{\sqrt{3}}
3
2
×
2
3
x
=
3
x
Therefore, Distance Between Centroid & vertex of equilateral triangle is \frac{x}{\sqrt{3}}
3
x