In triangle MNK, point F is the midpoint of side MN. It is known that the perimeter of the triangle FMK is 26

| August 17, 2021 | education

In triangle MNK, point F is the midpoint of side MN. It is known that the perimeter of the triangle FMK is 26 cm less than the perimeter of the triangle NKF. Find MK if NK = 51 cm.

Solution.

1. Perimeter is the sum of all sides. Perimeter of triangle FMK:

PFMK = FM + MK + FK.

2. Perimeter of triangle NKF:

PNKF = FN + NK + FK.

3. By condition, the perimeter of the triangle FMK is 26 cm less than the perimeter of the triangle NKF, ie:

PFMK + 26 = PNKF, i.e.

FM + MK + FK + 26 = FN + NK + FK.

4. FK in triangle FMK = FK in triangle NKF, because FK is a common side for triangles FMK and NKF. Then we transfer FK to one side of the equation:

FM + MK + 26 = FN + NK + FK – FK,

FM + MK + 26 = FN + NK.

5. FK is the median in the triangle MNK (by hypothesis, FK divides the side of MN in half), hence FM = FN, then we substitute FM instead of FN in the equation:

FM + MK + 26 = FM + NK.

Move FM to one side of the equation:

MK + 26 = FM + NK – FM,

MK + 26 = NK.

6. Find MK, if by condition NK = 51 cm:

MK + 26 = 51,

MK = 51 – 26,

MK = 25 (cm).

Answer: MK = 25 cm.



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