Step 1: Understanding the Concept:
The vertices of a triangle can be found if the mid-points of its sides are known.
Let mid-points be \( M_1, M_2, M_3 \). Then vertex \( A = M_1 + M_2 - M_3 \) and so on. Step 2: Key Formula or Approach:
Incentre \( I = \frac{ax_1 + bx_2 + cx_3}{a+b+c} \), where \( a, b, c \) are side lengths. Step 3: Detailed Explanation:
Vertices are:
\( A = (5/2 + 5/2 - 4, 3 + 7 - 5) = (1, 5) \).
\( B = (5/2 + 4 - 5/2, 3 + 5 - 7) = (4, 1) \).
\( C = (5/2 + 4 - 5/2, 7 + 5 - 3) = (4, 9) \).
Side lengths:
\( BC = \sqrt{(4-4)^2 + (9-1)^2} = 8 \).
\( AC = \sqrt{(4-1)^2 + (9-5)^2} = \sqrt{9+16} = 5 \).
\( AB = \sqrt{(4-1)^2 + (1-5)^2} = \sqrt{9+16} = 5 \).
Since \( AC = AB \), it is an isosceles triangle.
\( h = \frac{8(1) + 5(4) + 5(4)}{8+5+5} = \frac{48}{18} = \frac{8}{3} \).
\( k = \frac{8(5) + 5(1) + 5(9)}{18} = \frac{90}{18} = 5 \).
Value \( 3h + k = 3(8/3) + 5 = 8 + 5 = 13 \). Step 4: Final Answer:
The value of \( 3h + k \) is 13.
