Geometry Toolkit (DP IB Analysis & Approaches (AA): SL): Exam Questions

An artist has been commissioned to create a sculpture for the Mathematics department of a University. She decides to approximate a Fibonacci spiral from a 5 m length of copper wire by putting together a series of squares of increasing size with an arc of a quarter circle in each square. The centre of each arc is at a vertex of the square and the radius is the same as the side length of the square. The copper wire is to be used for the spiral and the edges of the outer rectangle only as shown in the diagrams below.

Calculate the length and width of the rectangle that encases the spiral.

In order to make the spiral stand out, various sections of the sculpture are to be filled in with coloured glass as shown below.

Calculate the area of glass that is required to complete the sculpture.

The points $\mathrm{K}\left(6,y\right)$ and $\mathrm{N}\left(x,9\right)$, where $x,y\in \mathrm{\mathbb{Z}}$, are such that $\mathrm{KN}=5$.

Find all the possible pairs of values of $x$ and $y$.

A new lamp has been designed that comprises an annulus containing the light bulbs on top of a stand in the shape of an equilateral triangle with side length 9.6 cm. The supporting edges of the stand are divided into thirds by the inner and outer edges of the light disc connecting to it at equally spaced points. The top vertex of the triangular base is located at the centre of the two circles that define the annulus.

[An annulus is a ring shaped object made up of a circle with a concentric circle removed from the centre].

A diagram of the lamp is shown below.

Calculate the area of the section of the annulus that can be seen from the front.

The points $\mathrm{P}\left(20,40\right)$ and $\mathrm{Q}\left(x,20\right)$ are such that $\mathrm{PQ}=25$.

Find the possible values of $x$.

It is given that $x<20$.

$\mathrm{R}$ is such that $\mathrm{Q}$ is the midpoint of $\left[\mathrm{PR}\right]$.

Find the coordinates of $\mathrm{R}$.

The points $\mathrm{A}\left(x,-5\right)$ and $\mathrm{B}\left(-1,y\right)$, where $x,y\in \mathrm{\mathbb{Z}}$, are such that $\mathrm{AB}=13$.

Find all the possible pairs of values of $x$ and $y$.

Three locations in a forest, $\mathrm{A}\left(3,5\right)$, $\mathrm{B}\left(-3,2\right)$ and $\mathrm{C}\left(6,-4\right)$, are marked out for an orienteering activity. These can be seen on the grid below. Each unit on the grid indicates a distance of 1 km.

(i) Find AC.

(ii) Find the bearing of A from C.

X is the midpoint of [BD]. X lies on [AC] such that AX : XC = 1 : 3.

Find the coordinates of D.

The logo of a new company comprises a circle of radius $r$ cm and centre O, with part of the interior area shaded. The diameter of the unshaded interior semicircle (the unshaded area below the dashed line in the diagram) is two thirds of the diameter of the larger circle. The remainder of the unshaded area is a sector of the main circle with a sector angle of 124°. This information is shown in the diagram below.

Show that the area of the shaded section is $\frac{13}{30}\pi r^2$ cm².

The sector angle of the unshaded sector is decreased.

Find the sector angle of the unshaded sector that is required to make the areas of the logo that are shaded and unshaded equal.

In the diagram below, ABCD shows a piece of geometric art on canvas measuring 58 cm by 78 cm. N is the midpoint of [BC] and M is the midpoint of [AB]. X is a point on [AC] such that AX : XC = 1 : 5. A straight line connects M to X. Y is the point where [AC] intersects [ND].

Calculate the area of the artwork that is painted black.

The piece of artwork is to be enlarged by a length scale factor of 6 and painted on an exterior wall of an art gallery. A 200 ml tin of paint costs $8 and covers an area of 2.4 m².

Calculate the cost of the paint that must be purchased to re-create the same black sections from part (a) on the wall.

The following diagram shows a circle with centre $\mathrm{O}$ and radius 2 cm. Points $\mathrm{A}$ and $\mathrm{B}$ lie on the circumference of the circle and $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=3\theta$, where $0<\theta <\frac{\pi }{3}$.

The tangents to the circle at $\mathrm{A}$ and $\mathrm{B}$ meet at the point $\mathrm{C}$.

Find the value of $\theta$ for which the shaded area is equal to the area of sector $\mathrm{OAB}$.