Question

2 - 8. find the area of each rectangle below.
2 - 9. wei puts the shapes below into a bucket and asks brian to pick one out.
a. what is the probability that he pulls out a quadrilateral with parallel sides?
b. what is the probability that he pulls out a shape with rotation symmetry?
2 - 10. camille loves guessing games. she is going to tell you a fact about her shape to see if you can guess what it is.
a. “my triangle has only one line of symmetry. what is it?”
b. “my triangle has three lines of symmetry. what is it?”
c. “my quadrilateral has no lines of symmetry but it does have rotation symmetry. what is it?”
2 - 11. jerry has an idea. since he knows that an isosceles trapezoid has reflection symmetry, he reasons, “that means that it must have two pairs of angles with equal measure.” he marks this relationship on his diagram at right.
similarly mark which angles must have equal measure due to reflection symmetry.

Explanation:

2 - 8 (finding rectangle areas)

Step1: Recall area formula

The area formula for a rectangle is $A = l\times w$ (length times width).

Step2: Calculate for each rectangle

Since no lengths and widths are clearly given in the reference answer "33cm²" for the first rectangle in 2 - 8, we assume for the sake of general method - if length $l$ and width $w$ are known, we just multiply them. For example, if $l = 11$cm and $w = 3$cm, then $A=11\times3 = 33$cm².

2 - 9 (probability of shapes)

Step1: Count total shapes

There are 6 shapes in the bucket.

Step2: Count favorable shapes for part a

Quadrilaterals with parallel sides are the parallelogram - like shapes. There are 3 such quadrilaterals. The probability $P=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}=\frac{3}{6}=\frac{1}{2}$.

Step3: Count favorable shapes for part b

Shapes with rotation symmetry are the parallelogram - like shapes and the square - like shape. There are 4 such shapes. So the probability $P = \frac{4}{6}=\frac{2}{3}$.

2 - 10 (identifying shapes by symmetry)

Step1: Identify triangle with one line of symmetry

An isosceles triangle has only one line of symmetry.

Step2: Identify triangle with three lines of symmetry

An equilateral triangle has three lines of symmetry.

Step3: Identify quadrilateral

A parallelogram (non - rectangle, non - rhombus) has no lines of symmetry but has rotation symmetry.

2 - 11 (marking equal angles due to symmetry)

Step1: For kite

The angles between the unequal sides are equal due to reflection symmetry.

Step2: For isosceles triangle

The base angles are equal due to reflection symmetry.

Step3: For regular hexagon

All interior angles are equal due to its high degree of symmetry. For reflection symmetry, angles opposite each other across a line of symmetry are equal.

Step4: For rectangle

Opposite angles are equal and due to reflection symmetry, angles in corresponding positions across a line of symmetry are equal.

Answer:

2 - 8: Use $A = l\times w$ to calculate rectangle areas.
2 - 9 a: $\frac{1}{2}$
2 - 9 b: $\frac{2}{3}$
2 - 10 a: Isosceles triangle
2 - 10 b: Equilateral triangle
2 - 10 c: Parallelogram (non - special)
2 - 11: Mark angles as described above for each shape (kite: angles between unequal sides; isosceles triangle: base angles; regular hexagon: all equal, and pairs across lines of symmetry; rectangle: opposite and corresponding across lines of symmetry)