Dissecting and forming shapes: Practice Questions

A large trapezium is formed by joining two identical shapes along their non-parallel equal sides. If the two shapes used are right-angled triangles, what is the most likely name of the resulting trapezium?

A large square with a side length of $$10 \text{ cm}$$ is dissected into two identical rectangles. Each of these rectangles is then dissected into two identical right-angled triangles. What is the area of one of these right-angled triangles?

A rectangular plot of land has an area of $$384 \text{ m}^2$$. It is divided into 3 equal square plots and a remaining rectangular plot. If the side length of each square plot is $$8 \text{ m}$$, what is the perimeter of the remaining rectangular plot?

A regular hexagon is dissected into several identical triangles by drawing lines from its center to every vertex. If the distance from the center to each vertex is 5 cm, and the side of the hexagon is also 5 cm, what specific type of triangles are formed?

A student has two identical isosceles trapeziums. Each has a top base of 4 cm, a bottom base of 8 cm, and a height of 3 cm. If they join them along their 8 cm bases, what polygon is formed, and how many sides does it have?

A square piece of paper is cut exactly in half along a straight line that connects the middle of two opposite sides. What two identical shapes are formed?

You have a rectangular piece of paper measuring $$ 10 \text{ cm} $$ by $$ 6 \text{ cm} $$. If you cut it into two identical squares and one smaller rectangle, what are the dimensions of the smaller rectangle?

A large square has a perimeter of $$ 48 \text{ cm} $$. It is cut into nine identical smaller squares. What is the area of one of these smaller squares?

A large square piece of paper has a side length of $$8 \text{ cm}$$.

a) The large square is first carefully cut into four identical smaller squares. What is the side length of each of these smaller squares?

b) Next, two of these smaller squares are each cut diagonally from one corner to the opposite corner. How many triangles are formed in total from these two squares? Describe the type of triangles formed (based on their sides and angles).

c) You now have all the pieces: the two small squares that were not cut, and all the triangles from part (b). Describe how you would arrange all these pieces to form a larger rectangle that is *not* a square. What would be the dimensions and the total area of this new rectangle?

A large rectangular piece of cardboard measures \(20\text{ cm}\) by \(15\text{ cm}\). A design is created by cutting out a square from one corner and a right-angled triangle from another corner. The square has a side length of \(5\text{ cm}\). The right-angled triangle has legs of \(10\text{ cm}\) and \(8\text{ cm}\).

a) What is the total area of the original rectangular cardboard?

b) Calculate the area of the square that was cut out.

c) Calculate the area of the right-angled triangle that was cut out.

d) The remaining cardboard, after the square and triangle are cut out, forms a new composite shape. Find the area of this remaining composite shape.

e) If the two cut-out shapes (the square and the triangle) are placed side-by-side, can they form a larger rectangle? Explain why or why not.