Question

Let \( A(1,2), B(3,4) \) and \( C(k,6) \) be three points such that the area of triangle \( ABC \) is 5 square units. The possible values of \( k \) are:

Hint:

Use the area formula for a triangle with coordinates and solve the resulting absolute value equation carefully to find possible values.

Answer 1

The area of a triangle formed by points \( A(x_1,y_1) \), \( B(x_2,y_2) \), and \( C(x_3,y_3) \) is given by: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Given: \[ A = (1,2), \quad B = (3,4), \quad C = (k,6) \] Area \( = 5 \) square units, so \[ 5 = \frac{1}{2} \left| 1(4 - 6) + 3(6 - 2) + k(2 - 4) \right| \] Calculate inside the absolute value: \[ = \frac{1}{2} \left| 1 \times (-2) + 3 \times 4 + k \times (-2) \right| = \frac{1}{2} \left| -2 + 12 - 2k \right| = \frac{1}{2} |10 - 2k| \] Multiply both sides by 2: \[ 10 = |10 - 2k| \] This gives two cases: Case 1: \[ 10 - 2k = 10 \implies -2k = 0 \implies k = 0 \] Case 2: \[ 10 - 2k = -10 \implies -2k = -20 \implies k = 10 \] But \( k = 10 \) is not in the options. Double-check the calculation carefully. Re-examining: \[ 10 = |10 - 2k| \implies |10 - 2k| = 10 \] Possible values: \[ 10 - 2k = 10 \implies k = 0 \] or \[ 10 - 2k = -10 \implies -2k = -20 \implies k = 10 \]