Problem
A rectangle's diagonals are equal and bisect at the crossing, so each piece to a corner is half of 32: 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 SubproblemsThe top angle FPE is 120 degrees, and its neighbor along the straight line fills the rest: angle FPC = 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 DiagramWith PF = PC = 16 cm and apex FPC = 60 degrees, the two base angles are also 60 degrees, so triangle FPC is equilateral and FC = 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!