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State whether the given statement is true or false.

The vertices of a $Δ A B C$ are $A (7, 8), B (4, 2)$ areand $C (8, 2)$ and. If the midpoint of the side BC is (6,2), then the median AD divides the $ΔABC$ into two triangles equal in area.

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True

False

answer is A.

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Detailed Solution

We have been given,
If the midpoint of the side BC is (6,2), then the median AD divides the

△ A B C

into two triangles equal in area.
The vertices of the triangle are A, B, and C.
D is the midpoint of BC.
Coordinates of A, B, C and D are as following,

A ≡ (7, 8) B = (4, 2) C = (8, 2) D = (6, 2)

Now, calculate the area of triangles,

ar (ΔABD) = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

Triangle ABD,
Substituting

x_{1} = 7, x_{2} = 4, x_{3} = 6, y_{1} = 8, y_{2} = 2, and y_{3} = 2

, in the formula,

ar (ΔABD) = \frac{1}{2} |7 (2 - 2) + 4 (2 - 8) + 6 (8 - 2)|

ar (ΔABD) = \frac{1}{2} |12| = 6 sq units

Triangle ACD,

ar (ΔACD) = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

Substituting

x_{1} = 7, x_{2} = 8, x_{3} = 6, y_{1} = 8, y_{2} = 2, and y_{3} = 2

, in the formula,

ar (ΔACD) = \frac{1}{2} |7 (2 - 2) + 8 (2 - 8) + 6 (8 - 2)|

ar (ΔACD) = \frac{1}{2} | 0 - 48 + 36 |

ar (ΔACD) = \frac{1}{2} | - 12 |

ar (ΔACD) = 6 sq . unit

\Rightarrow ar (ΔABD) = ar (ΔACD)

For confirmation, calculating the area of triangle ABC.

ar (ΔABC) = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

Substituting

x_{1} = 7, x_{2} = 4, x_{3} = 8, y_{1} = 8, y_{2} = 2, and y_{3} = 2

, in the formula,

ar (ΔABC) = \frac{1}{2} |7 (2 - 2) + 4 (2 - 8) + 8 (8 - 2)|

ar (ΔABC) = \frac{1}{2} | 0 - 24 + 48 |

ar (ΔABC) = \frac{1}{2} | 24 |

ar (ΔABC) = 12 sq . unit

\Rightarrow ar (ΔABC) = ar (ΔACD) + ar (ΔABD)

Therefore, our statement is true.
Hence, option 1 is correct.

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State whether the given statement is true or false.The vertices of a ΔABC are A(7,8),B(4,2) areand C(8,2) and. If the midpoint of the side BC is (6,2), then the median AD divides the ΔABC into two triangles equal in area.

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State whether the given statement is true or false.

The vertices of a $Δ A B C$ are $A (7, 8), B (4, 2)$ areand $C (8, 2)$ and. If the midpoint of the side BC is (6,2), then the median AD divides the $ΔABC$ into two triangles equal in area.