Question

In an isosceles triangle,length of congruent side is 13 cm and its base is 10cm. Find the distance between the vertex opposite the base and the centroid.

Answer 1

Final Answer : 8cm

Steps and Understanding :
1) Given :
ABC be an isosceles triangle whose congruent sides are AC and BC.
AC=BC=13cm
AB = 10cm.

2) We will mark all points by taking origin as A (0,0) in first Quadrant of Cartesian Co-ordinate system.
Then,
A= (0,0)
B = (10,0)

Now,
I isosceles triangle,
CE is median passing through centroid D
Therefore,
AE = BE = 5cm
=> E = (5,0)

3) In triangle ACE,
By Pythagoras Theorem,
EC^2. = AC^2 - AE^2
=> EC^2 = 13^2 - 5^2
=> EC = 12cm
=> C = (5,12)

4) Centroid of triangle ABC,
C =( ( x1 + x2 + x3)/ 3 ,(y1+y2+y3)/3 )
=> C =((0+ 10+ 5)/3 ,(0+0+12)/3
=> C = (15/3 , 12/3 )
=> C = (5, 4)

5) Required :
By distance formula,
Distance between vertex opposite the base and controid is :
CD =
$cd = \sqrt{ {(5 - 5)}^{2} + {(12 - 4)}^{2} } \\ = > cd \: = 8cm$


Hence,
Distance between the vertex opposite the base and the controid is 8cm.

Answer 2

In the attachments I have answered this problem.

Pythagoras theorem is applied to find the length of the median AD.

It is clear that
The centroid G of a triangle divides the median AD in the ratio 2:1 .

Hence the distance AG is calculated as 8cm

I hope this answer helps you