2D shapes & transformations

Fluency

Reasoning

$x = 60^\circ$ . Method: $x + 2x + 100 + 80 = 360$ ; so $3x = 180$ .
$5$ cm. Method: diagonals meet at right angles and bisect each other; half-diagonals are $3$ and $4$ ; Pythagoras $\sqrt{3^2 + 4^2}$ (or use the $3$ - $4$ - $5$ triangle).
$100^\circ$ . Method: base angles are both $40^\circ$ ; apex $= 180 - 80$ .
$26$ cm. Method: $5 + 5 + 8 + 8$ .
$5$ lines of symmetry.
$(-3, -3)$ . Method: reflect gives $(-3, -2)$ ; translate gives $(-3, -3)$ .
Reflection in the $y$ -axis.
$60^\circ$ . Method: $2x + 240 = 360$ , so $x = 60$ .

Reasoning

Yes, Ida is correct. A rectangle is a quadrilateral with four right angles; a square meets this (and also has all sides equal), so every square is a rectangle. The extra property “all sides equal” just makes the square a special rectangle.
No. A rectangle needs only four right angles; a square also needs four equal sides. A $3 \times 5$ rectangle has four right angles but unequal sides, so it is a rectangle but not a square.
A trapezium has at least one pair of parallel sides; a parallelogram has two pairs. Under the broad (inclusive) definition, every parallelogram is a trapezium, so the student’s claim is false. Under the “exactly one pair” definition, a parallelogram is not a trapezium, so the student is correct. Both definitions are used in textbooks.
A $180^\circ$ rotation about the origin is a half-turn: each point moves to the point on the opposite side of the origin, the same distance away. Flipping direction from the origin negates both coordinates, so $(a, b) \to (-a, -b)$ .

Problem solving

$6$ m total. Method: rectangle perimeter $= 2(1.2 + 1.8) = 6$ m; minus $1.2$ m (the top edge is shared) $= 4.8$ ; plus the two $1.2$ m slanted sides of the equilateral triangle; total $= 4.8 + 2.4 = 7.2$ m. Correct answer: $7.2$ m.
$30$ cm. Method: $2(6 + 9)$ .
$26$ cm. Method: $10 + 6 + 5 + 5$ .
$22$ cm. Method: $4 + 4 + 7 + 7$ .
$(1, -1)$ . Method: rotate $(1, 1)$ by $-90^\circ$ gives $(1, -1)$ ; reflect in $x$ -axis gives $(1, 1)$ . Wait - rotating $(1,1)$ by $-90^\circ$ (clockwise) gives $(1, -1)$ . Then reflecting in the $x$ -axis flips the $y$ -coordinate, giving $(1, 1)$ . Final image: $(1, 1)$ .

Tier 1: basic skills