Problem
The diagonals of a rectangle are equal and cut each other in half at the crossing point. Since each diagonal is 32 cm, every piece from P out to a corner is half of that: PF = PC = 16 cm.
A rectangle's diagonals are equal and bisect each other, so all four half-pieces are the same 16 cm.
4.G.A.2Identify SubproblemsWhere the diagonals cross, opposite angles are equal and neighboring angles on the straight line add to 180 degrees. The top angle FPE is 120 degrees, so the angle FPC sitting against side FC is 180 - 120 = 60 degrees.
Angles around the crossing fill a straight line, so the neighbor of 120 degrees is 60 degrees.
4.MD.C.7Draw A DiagramTriangle FPC has PF = PC = 16 cm, so it is isosceles and its two base angles are equal. The top angle FPC is 60 degrees, leaving 180 - 60 = 120 degrees to share equally, so each base angle is 60 degrees. All three angles are 60 degrees, so the triangle is equilateral and FC = PF = PC = 16 cm.
An isosceles triangle with a 60 degree apex has all angles 60 degrees, so every side matches: FC equals the 16 cm half-diagonal.
4.G.A.2Identify SubproblemsTriangle FPC is equilateral, so its third side FC is 16 cm, the same length as the two half-diagonals PF and PC.
When a triangle's three angles are all equal you can fold it so any two corners land on each other, and the sides meeting those corners match, so all three sides come out equal and FC becomes 16 cm like PF and PC.
Every angle inside triangle FPC measures 60 degrees.
The top angle FPC is 60 degrees, because it sits right next to the marked 120 degree angle along the straight diagonal, and side by side those two angles fill a straight line.
The two bottom angles are equal to each other, because the sides sloping up to P, namely PF and PC, are the same length, so folding along the line through P lays one bottom angle exactly onto the other.
Each bottom angle is also 60 degrees, because the three angles add to 180, the top angle already uses 60, and the remaining 120 splits evenly between the two equal bottom angles.
The square ABCF has side FC = 16 cm, and a square has four equal sides, so its perimeter is 4 x 16 = 64 cm.
Perimeter of a square is four equal sides added up, which is just 4 times one side.
4.MD.A.3Identify SubproblemsWhen a 16-and-16 isosceles triangle has a 60 degree tip, it must be equilateral, so the third side is 16 too -- pure Grade 4 angle sense!