In an Isosceles Triangle, Length of the Congruent Sides is 13 Cm and Its Base is 10 Cm. Find the Distance Between the Vertex Opposite the Base and the Centroid. - Geometry Mathematics 2

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

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Solution

\[\text{Area of the triangle} = \sqrt{s\left( s - a \right)\left( s - b \right)\left( s - c \right)}\]
\[s = \frac{a + b + c}{2}\]
\[ = \frac{13 + 13 + 10}{2}\]
\[ = \frac{36}{2}\]
\[ = 18 cm\]
\[\text{Area of the triangle} = \sqrt{18\left( 18 - 13 \right)\left( 18 - 13 \right)\left( 18 - 10 \right)}\]
\[ = \sqrt{2 \times 3 \times 3 \times 5 \times 5 \times 2 \times 2 \times 2}\]
\[ = 60 sq . cm\]
\[\text{Also}, \]
\[\text{Area of the triangle} = \frac{1}{2} \times base \times height\]
\[ \Rightarrow 60 = \frac{1}{2} \times 10 \times \text{height}\]
\[ \Rightarrow \text{height} = \frac{60}{5}\]
\[ \Rightarrow \text{height} = 12 cm\]

The centroid is located two third of the distance from any vertex of the triangle.

\[\therefore \text{Distance between the vertex and the centroid} = \frac{2}{3} \times 12 = 8 cm\]

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

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Chapter 2: Pythagoras Theorem - Problem Set 2 [Page 44]

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Balbharati Mathematics 2 Geometry 10th Standard SSC Maharashtra State Board
Chapter 2 Pythagoras Theorem
Problem Set 2 | Q 14 | Page 44

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