Isosceles triangle from a rectangle diagonal

4.G.A.24.MD.C.7 · adapt · grade 4

Archetype: Isosceles and Equilateral Angle Chaining · step in a 6-type progression

Square $ABCF$ and rectangle $FCDE$ are joined edge to edge without overlapping. If one diagonal of rectangle $FCDE$ is $32 \text{ cm}$ , find the perimeter of square $ABCF$ .

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Understand

A square ABCF and a rectangle FCDE sit side by side, glued along the shared vertical side FC. The rectangle's two diagonals are drawn and cross at P. The angle at P (the one opening toward the top) is 120 degrees, and each diagonal of the rectangle is 32 cm. I need the perimeter of the square.

Givens

Square ABCF is on the left; rectangle FCDE is on the right; they share side FC.
Both diagonals of rectangle FCDE are drawn and meet at point P.
One diagonal of the rectangle measures 32 cm.
The angle at P opening toward the top (angle FPE) is 120 degrees.

Unknowns

The perimeter of square ABCF.

Constraints

In a rectangle the two diagonals are equal and bisect each other, so the four pieces from P to the corners are all equal.
FC is both the left side of the rectangle and a side of the square, so the square's side length equals FC.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram#6 Guess and Check

Split the work: first use the rectangle's diagonal properties to find the lengths PF and PC and the angle inside triangle FPC, then recognize triangle FPC as a special triangle that gives FC, then multiply by 4 for the square's perimeter.

Execute

#7 Identify Subproblems 4.G.A.2

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.

32 \div 2 = 16 \text{ cm}

A rectangle's diagonals are equal and bisect each other, so all four half-pieces are the same 16 cm.

#1 Draw a Diagram 4.MD.C.7

Where the diagonals cross they make two pairs of opposite angles that sum to 180 degrees. The top angle FPE is 120 degrees, so the angle FPC sitting against side FC is 180 - 120 = 60 degrees.

180^\circ - 120^\circ = 60^\circ

Angles around the crossing fill a straight line, so the neighbor of 120 degrees is 60 degrees.

#7 Identify Subproblems 4.G.A.2

Triangle 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.

(180^\circ - 60^\circ) \div 2 = 60^\circ \Rightarrow FC = 16 \text{ 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.

#7 Identify Subproblems 4.MD.A.3

The square ABCF has side FC = 16 cm, and a square has four equal sides, so its perimeter is 4 x 16 = 64 cm.

4 \times 16 = 64 \text{ cm}

Perimeter of a square is four equal sides added up, which is just 4 times one side.

Answer: 64 cm

Review

FC = 16 cm is exactly half of the 32 cm diagonal, which is believable, and the square's perimeter 64 cm = 4 x 16 has the right size and unit (centimeters). The equilateral triangle nicely matches the 60 degree angle the 120 degree mark forces.

Guess and check (tool 6): if the side were larger than 16 cm the angle at P facing FC would exceed 60 degrees and its top neighbor would drop below 120 degrees; only FC = 16 cm produces the given 120 degrees, confirming the answer.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using rectangle and square properties (equal diagonals, equal bisected halves, equal sides) and identifying the equilateral triangle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding angle FPC = 60 degrees from the 120 degree mark and splitting the triangle's remaining angle.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Computing the perimeter of the square as 4 times its side length.

💡 When 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!