GeometryIB MYP Maths Extended: Topic test

1. 1
   A regular polygon has an interior angle of $156^\circ$.

   (a)

   What is the size of one exterior angle of the polygon?

   [1 mark]

   • A$24^\circ$
   • B$204^\circ$
   • C$12^\circ$
   • D$66^\circ$

   (b)

   How many sides does the polygon have?

   [1 mark]

   • A$12$
   • B$14$
   • C$15$
   • D$16$

   (c)

   Write down a formula for the sum of the interior angles of an $n$-sided polygon. Use it to find the sum of the interior angles of this polygon.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A triangular sail has side lengths $7$ m, $24$ m and $25$ m.

   (a)

   Which statement proves that the sail is a right-angled triangle?

   [1 mark]

   • A$7+24=31$, which is greater than $25$.
   • B$7\times24=168$, which is not equal to $25^2$.
   • C$25-24=1$, which is less than $7$.
   • D$7^2+24^2=625$ and $25^2=625$.

   (b)

   What is the area of the sail in m$^2$?

   [1 mark]

   • A$168$
   • B$84$
   • C$87.5$
   • D$175$

   (c)

   A second sail has side lengths $8$ m, $15$ m and $18$ m. Show whether this triangle is acute, right-angled or obtuse.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A craft company in Kyoto makes shapes in two sizes. The small flag has a base of $30$ cm and an area of $360$ cm$^2$, and the large flag is mathematically similar to it, with a base of $75$ cm. The company also sells a small gift box that is $12$ cm tall and holds $0.9$ litres, and a large gift box that is mathematically similar to it and $20$ cm tall.

   (a)

   Find the area of the large flag.

   [3 marks]

   (b)

   Find the volume of the large gift box in litres, to $3$ significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The points $A$, $B$, $C$ and $D$ lie in that order on a circle with centre $O$. Angle $ABC=(3x+15)^\circ$ and angle $ADC=(2x+25)^\circ$.

   (a)

   Find $x$ and the size of angle $ABC$. Then find the size of the non-reflex angle $AOC$. Give a reason for each step.

   [6 marks]

   (b)

   The tangents to the circle at $A$ and at $C$ meet at the point $T$. Find angle $ATC$ and angle $TAC$, giving a reason for each step.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Triangle $T$ has vertices $P(1,2)$, $Q(5,2)$ and $R(5,4)$ on a coordinate grid.

   (a)

   The point $R$ is reflected in the line $y=x$. What are the coordinates of its image?

   [1 mark]

   • A$(5,-4)$
   • B$(4,5)$
   • C$(-4,-5)$
   • D$(-5,-4)$

   (b)

   Triangle $T$ is enlarged with scale factor $3$ and centre the origin. What are the coordinates of the image of $R$?

   [1 mark]

   • A$(8,7)$
   • B$(5,12)$
   • C$(15,4)$
   • D$(15,12)$

   (c)

   Triangle $T$ is mapped by a single transformation onto the triangle with vertices $(1,-2)$, $(5,-2)$ and $(5,-4)$. Describe the transformation fully.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=\begin{pmatrix}2\\-3\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}4\\1\end{pmatrix}$.

   (a)

   Find $\mathbf{a}+\mathbf{b}$.

   [1 mark]

   • A$\begin{pmatrix}8\\-3\end{pmatrix}$
   • B$\begin{pmatrix}2\\4\end{pmatrix}$
   • C$\begin{pmatrix}6\\-2\end{pmatrix}$
   • D$\begin{pmatrix}6\\-4\end{pmatrix}$

   (b)

   Find the magnitude of $\mathbf{b}$.

   [1 mark]

   • A$\sqrt{17}$
   • B$5$
   • C$17$
   • D$3$

   (c)

   Find $2\mathbf{a}-\mathbf{b}$ as a column vector.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The matrices $\mathbf{P}$ and $\mathbf{Q}$ are given by $\mathbf{P}=\begin{pmatrix}1&3\\2&-1\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}0&2\\4&1\end{pmatrix}$.

   (a)

   Calculate $\mathbf{PQ}$.

   [3 marks]

   (b)

   Calculate $\mathbf{QP}$ and state whether $\mathbf{PQ}=\mathbf{QP}$. Give a reason.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A grain silo in Iowa is made from a cylinder of radius $3$ m and height $8$ m, topped by a hemisphere of radius $3$ m. The base of the silo is on the ground and is not painted. The volume of a sphere is $\frac{4}{3}\pi r^{3}$, its surface area is $4\pi r^{2}$, the volume of a cylinder is $\pi r^{2}h$ and its curved surface area is $2\pi rh$. Use the $\pi$ button on your calculator.

   (a)

   Find (i) the total volume of the silo, and (ii) the total area to be painted. Give each answer to $3$ significant figures. [3 marks each]

   [6 marks]

   (b)

   A model of the silo is made to a scale of $1:20$. Find the volume of the model in litres, and the area of the model that needs painting in m$^2$, each to $3$ significant figures. ($1$ m$^3=1000$ litres.)

   [6 marks]

   Total for question 8: 12 marks

End of questions