Question

in an isosceles triangle length of the congruent side is 13 cm and its base is 10 cm find distance between the vertex opposite to the base and the centroid​

Answer 1

HERE IS YOUR ANSWER

The centroid is located two third of the distance from any vertex of the triangle. Hence, the distance between the vertex opposite the base and the centroid is 8 cm.

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Answer 2

Answer:

The distance between the vertex opposite the base and centroid is 8 cm.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure,

△ABC is an isosceles triangle.

∴ AB = AC = 13 cm

BC = 10 cm

Construction:

Draw seg AM ⊥ side BC such that B - M - C.

Now,

In △AMB & △AMC,

∠AMB = ∠AMC = 90° - - [ Construction ]

Seg AB ≅ seg AC - - [ Given ]

Seg AM ≅ seg AM - - [ Common side ]

∴ △AMB ≅ △AMC - - [ Hypotenuse - side test ]

⇒ Seg BM ≅ seg CM - - ( 1 ) [ c.s.c.t. ]

Now,

BM = CM = ½ BC

⇒ BM = CM = ½ × 10

⇒ BM = CM = 5 cm

Now,

In △AMB,

∠AMB = 90° - - [ Construction ]

∴ AB² = AM² + BM² - - [ Pythagors theorem ]

⇒ ( 13 )² = AM² + ( 5 )²

⇒ AM² = ( 13 )² - 5²

⇒ AM² = 169 - 25

⇒ AM² = 144

⇒ AM = 12 cm - - [ Taking square roots ]

Now,

BM = CM - - [ From ( 1 ) ]

∴ Point M is midpoint of BC.

∴ Seg AM is median.

∴ Point G is the centroid.

Now,

By the property of centroid of triangle,

AG = ⅔ AM

⇒ AG = ⅔ × 12

⇒ AG = 2 × 12 ÷ 3

⇒ AG = 2 × 4

⇒ AG = 8 cm

∴ The distance between the vertex opposite the base and centroid is 8 cm.