Teach Me Maths

Properties of 2D Shapes

Question 11 mark

What is the mathematical name of this polygon?

Hint

Count its straight sides.

Worked solution

What is the mathematical name of this polygon?

  1. Hexagon
  2. Quadrilateral
  3. Pentagon
  4. Octagon
  1. The polygon has five sides, so it is a pentagon.

Question 21 mark

The triangle has matching marks on all three sides. What type of triangle is it?

Hint

Matching marks mean equal lengths.

Worked solution

The triangle has matching marks on all three sides. What type of triangle is it?

  1. Equilateral
  2. Scalene
  3. Right-angled
  4. Obtuse-angled
  1. All three sides are equal, so the triangle is equilateral.

Question 31 mark

The square mark is shown on the triangle. Which description must be correct?

Hint

The small square marks an angle of 90°.

Worked solution

The square mark is shown on the triangle. Which description must be correct?

  1. It is equilateral.
  2. It is right-angled.
  3. All of its angles are acute.
  4. It has three equal sides.
  1. The triangle contains a right angle, so it is right-angled. The mark does not tell us that any sides are equal.

Question 41 mark

The quadrilateral has exactly one pair of parallel sides. What is its name?

Hint

Use the parallel-side property.

Worked solution

The quadrilateral has exactly one pair of parallel sides. What is its name?

  1. Parallelogram
  2. Rectangle
  3. Kite
  4. Trapezium
  1. A quadrilateral with exactly one pair of parallel sides is a trapezium.

Question 51 mark

All four sides are equal and none of the angles is a right angle. What is the most specific name of the quadrilateral?

Hint

A square needs right angles as well as four equal sides.

Worked solution

All four sides are equal and none of the angles is a right angle. What is the most specific name of the quadrilateral?

  1. Square
  2. Rhombus
  3. Rectangle
  4. Trapezium
  1. Four equal sides make a rhombus. With no right angles, this one is not a square.

Question 61 mark

The marks show two different pairs of adjacent equal sides. What is the name of this quadrilateral?

Hint

Adjacent sides meet at a vertex.

Worked solution

The marks show two different pairs of adjacent equal sides. What is the name of this quadrilateral?

  1. Rectangle
  2. Regular quadrilateral
  3. Kite
  4. Parallelogram
  1. The two upper sides form one equal pair and the two lower sides form another. This is a kite.

Question 71 mark

Which statement describes every regular polygon?

Hint

Regular describes both the sides and the angles.

Worked solution

Which statement describes every regular polygon?

  1. Only its side lengths must be equal.
  2. All its side lengths and all its interior angles are equal.
  3. It must have four sides.
  4. All its angles must be right angles.
  1. A regular polygon has all side lengths equal and all interior angles equal. Equal sides alone are not enough, as a non-square rhombus shows.

Question 81 mark

ABCD is a parallelogram. Work out the length x, in centimetres.

Hint

Opposite sides of a parallelogram are equal.

Worked solution

ABCD is a parallelogram. Work out the length x, in centimetres.

  1. The side opposite x is 9.3 cm, so x=9.3 cm.

Question 91 mark

How many right angles does every rectangle have?

Hint

Use the definition of a rectangle.

Worked solution

How many right angles does every rectangle have?

  1. A rectangle has four right angles. Its adjacent sides do not have to be equal.

Question 101 mark

A triangle has side lengths 5 cm, 7 cm and 9 cm. What type of triangle is it when classified by its side lengths?

Hint

Compare the three side lengths.

Worked solution

A triangle has side lengths 5 cm, 7 cm and 9 cm. What type of triangle is it when classified by its side lengths?

  1. Scalene
  2. Equilateral
  3. Isosceles
  4. Regular
  1. All three lengths are different, so the triangle is scalene.

Question 111 mark

A quadrilateral has two pairs of parallel sides. One of its angles is a right angle. Its sides are not all the same length. What is the mathematical name of this quadrilateral?

Hint

Two pairs of parallel sides make a parallelogram. What does one right angle tell you about the other three angles?

Worked solution

A quadrilateral has two pairs of parallel sides. One of its angles is a right angle. Its sides are not all the same length. What is the mathematical name of this quadrilateral?

  1. Square
  2. Rhombus
  3. Rectangle
  4. Kite
  1. Two pairs of parallel sides make it a parallelogram. In a parallelogram, angles next to each other add up to 180° and opposite angles are equal, so one right angle makes all four angles right angles.
  2. A quadrilateral with four right angles is a rectangle or a square. Its sides are not all the same length, so it is a rectangle.

Question 121 mark

A student says that a rectangle cannot be a parallelogram. Which statement explains why the student is wrong?

Hint

A shape can belong to a broad family and also have a more specific name.

Worked solution

A student says that a rectangle cannot be a parallelogram. Which statement explains why the student is wrong?

  1. A rectangle has both pairs of opposite sides parallel.
  2. Every parallelogram has four right angles.
  3. Every rectangle has four equal sides.
  4. A rectangle has exactly one pair of parallel sides.
  1. Every rectangle has both pairs of opposite sides parallel, which satisfies the definition of a parallelogram.
  2. Having four right angles makes a rectangle a special type of parallelogram.

Question 132 marks

ABCD is a parallelogram. A is (1, 1), B is (6, 1) and C is (8, 4). Find the coordinates of D.

Write your answer as (x, y)

Hint

Opposite sides are parallel and equal in length. Compare the move from B to C with the move from A to D.

Worked solution

ABCD is a parallelogram. A is (1, 1), B is (6, 1) and C is (8, 4). Find the coordinates of D.

  1. From B to C the move is 2 units right and 3 units up.
  2. The same move from A gives D = (1 + 2, 1 + 3) = (3, 4).
  3. The completed diagram shows the result.
    Shape on coordinate gridA is (1,1), B is (6,1), and C is (8,4). Segments AB and BC are drawn. D is (3,4), completing the parallelogram.ABCD-22468-4-22460xyEach square is 1 unit
    Completed parallelogram

Question 142 marks

The diagonals of parallelogram ABCD meet at M. AM is 4.5 cm. Work out AC, in centimetres.

Hint

The diagonals of a parallelogram bisect each other.

Worked solution

The diagonals of parallelogram ABCD meet at M. AM is 4.5 cm. Work out AC, in centimetres.

  1. AM=MC=4.5 cm.
  2. AC=4.5+4.5=9 cm.

Question 152 marks

The diagram shows a parallelogram. Work out x.

Hint

Use the equal opposite sides to form an equation.

Worked solution

The diagram shows a parallelogram. Work out x.

  1. 2x+1=15.
  2. 2x=14, so x=7.

Question 16Challenge4 marks

The diagram shows a kite. Its two upper sides are each (2x+3) cm long and its two lower sides are each (3x-1) cm long. The perimeter of the kite is 74 cm. Work out the length of one of the longer sides, in centimetres.

Hint

Use both pairs of equal sides when forming the perimeter equation.

Worked solution

The diagram shows a kite. Its two upper sides are each (2x+3) cm long and its two lower sides are each (3x-1) cm long. The perimeter of the kite is 74 cm. Work out the length of one of the longer sides, in centimetres.

  1. 2(2x+3)+2(3x-1)=74.
  2. 10x+4=74, so x=7.
  3. The two lengths are 2\times7+3=17 cm and 3\times7-1=20 cm. The longer side is 20 cm.

Question 17Challenge3 marks

Two adjacent angles of a parallelogram are (3x + 10)° and (5x + 18)°. Work out the smaller of these two angles, in degrees.

Hint

Adjacent angles in a parallelogram add to 180°. Finding x is not the final answer.

Worked solution

Two adjacent angles of a parallelogram are (3x + 10)° and (5x + 18)°. Work out the smaller of these two angles, in degrees.

  1. (3x+10)+(5x+18)=180.
  2. 8x+28=180, so x=19.
  3. The angles are 3\times19+10=67^\circ and 5\times19+18=113^\circ. The smaller is 67°.

Question 18Challenge4 marks

ABCD is a rectangle. Its diagonals meet at M. Work out the length AC, in centimetres.

Hint

The diagonals of a rectangle are equal and bisect each other.

Worked solution

ABCD is a rectangle. Its diagonals meet at M. Work out the length AC, in centimetres.

  1. AC=2AM=2(2x+3) and AC=BD=5x-4.
  2. 4x+6=5x-4, so x=10.
  3. AC=5\times10-4=46 cm.

Question 19Challenge3 marks

A is (−2, 1), B is (1, 5) and C is (4, 1). D has y-coordinate −2. ABCD is a kite with AD = CD. Give the coordinates of D.

Write your answer as (x, y)

Hint

D must be the same distance from A and C, which lie on a horizontal line.

Worked solution

A is (−2, 1), B is (1, 5) and C is (4, 1). D has y-coordinate −2. ABCD is a kite with AD = CD. Give the coordinates of D.

  1. The midpoint of AC has x-coordinate (−2 + 4) ÷ 2 = 1.
  2. Points equally distant from A and C lie on the vertical line through this midpoint, so D has x-coordinate 1.
  3. Its given y-coordinate is −2, so D = (1, −2).
  4. The completed diagram shows the result.
    Shape on coordinate gridA is (-2,1), B is (1,5), and C is (4,1). AB and BC are drawn. D is (1,-2), completing the kite.ABCD-22468-4-2246xyEach square is 1 unit
    Completed kite

Question 20Challenge1 mark

A quadrilateral has exactly one line of symmetry. None of its sides are parallel. What is the mathematical name of this quadrilateral?

Hint

Which of these shapes have parallel sides? Then check the number of lines of symmetry.

Worked solution

A quadrilateral has exactly one line of symmetry. None of its sides are parallel. What is the mathematical name of this quadrilateral?

  1. Isosceles trapezium
  2. Kite
  3. Rhombus
  4. Parallelogram
  1. An isosceles trapezium has a pair of parallel sides, and so do a rhombus and a parallelogram, so none of these fit.
  2. A kite has no parallel sides and exactly one line of symmetry, through the corners where its equal sides meet. The quadrilateral is a kite.

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