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A, B, C are respectively the points (1,2...

A, B, C are respectively the points (1,2), (4, 2), (4, 5). If `T_(1), T_(2)` are the points of trisection of the line segment BC, the area of the Triangle `A T_(1) T_(2)` is

A

1

B

`(3)/(2)`

C

2

D

`(5)/(2)`

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To find the area of triangle \( A T_1 T_2 \) where \( A, B, C \) are the points \( (1, 2), (4, 2), (4, 5) \) and \( T_1, T_2 \) are the points of trisection of line segment \( BC \), we will follow these steps: ### Step 1: Identify Points The points are given as: - \( A(1, 2) \) - \( B(4, 2) \) - \( C(4, 5) \) ### Step 2: Find Points of Trisection \( T_1 \) and \( T_2 \) To find the points of trisection \( T_1 \) and \( T_2 \) of the line segment \( BC \): - The coordinates of \( B \) are \( (4, 2) \) and \( C \) are \( (4, 5) \). - The distance between \( B \) and \( C \) is \( 5 - 2 = 3 \). - Since \( T_1 \) and \( T_2 \) trisect the segment, they divide it into three equal parts of \( 1 \) unit each. Thus: - \( T_1 \) is at \( (4, 2 + 1) = (4, 3) \) - \( T_2 \) is at \( (4, 2 + 2) = (4, 4) \) ### Step 3: Calculate the Area of Triangle \( A T_1 T_2 \) The area of a triangle given vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Substituting the coordinates: - \( A(1, 2) \) - \( T_1(4, 3) \) - \( T_2(4, 4) \) The area becomes: \[ \text{Area} = \frac{1}{2} \left| 1(3 - 4) + 4(4 - 2) + 4(2 - 3) \right| \] Calculating each term: \[ = \frac{1}{2} \left| 1(-1) + 4(2) + 4(-1) \right| \] \[ = \frac{1}{2} \left| -1 + 8 - 4 \right| \] \[ = \frac{1}{2} \left| 3 \right| = \frac{3}{2} \] ### Step 4: Final Area Calculation Thus, the area of triangle \( A T_1 T_2 \) is \( \frac{3}{2} \) square units. ### Summary The area of triangle \( A T_1 T_2 \) is \( \frac{3}{2} \) square units.
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