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

Point $\mathrm{A}$ has coordinates $\left(4,-6\right)$ and point $\mathrm{B}$ has coordinates $\left(8,6\right)$.

Find $\mathrm{AB}$.

Find the equation of the line through $\mathrm{A}$ and $\mathrm{B}$.

Give your answer in the form $y=mx+c$.

$\mathrm{M}$ is the midpoint of $\left[\mathrm{AB}\right]$.

(i) Find the coordinates of $\mathrm{M}$.

Line $L$ is perpendicular to $\left[\mathrm{AB}\right]$ and passes through $\mathrm{M}$.

(ii) Find the equation of $L$. Give your answer in the form $y=mx+c$.

The diagram below shows a circle with a 68° sector cut from it. The radius of the circle is 5 cm.

Find the length of

(i) the minor arc AB

(ii) the major arc AB.

Find the area of the shaded region.

A lawn sprinkler sprays water over a lawn, turning through an angle of 160° with a maximum spray distance of $x$ m, as shown in the diagram below. The lawn sprinkler waters 20 m² of the lawn.

Calculate the value of $x$.

Calculate the length of the outer arc.

A windscreen wiper blade is 0.8 m long. When in motion, the blade rotates through an angle of $\theta$° and wipes an area of $\frac{4}{15}\pi$ m².

Calculate the value of $\theta$.

Calculate the length travelled by the outer edge of the blade.

The diagram below shows a dirt racetrack where the straights are 426 m long and the longest distance from one end of the track to the other is 554 m.

Find the total distance around the racetrack.

Find the total area enclosed by the racetrack.

The diagram below shows a cookie cutter in the shape of a heart, constructed from a triangle and two identical semicircles. The height of the triangle is 8 cm and its base is $\mathrm{AB}=13.34$ cm.

Find $\mathrm{AC}$.

Calculate the total area of the heart.

Bob makes some cookie dough and rolls it out on his kitchen bench. The cookie dough covers 1314 cm².

Find the number of full cookies Bob can cut from the dough.

The diagram below shows a slice of pizza that forms a sector of a circle with angle 60° and radius 15 cm. The width of the crust is 2.2 cm.

Find the perimeter of the slice of pizza.

Find the area of the crust.

Point $\mathrm{A}$ has coordinates $\left(-1,7\right)$ and point $\mathrm{B}$ has coordinates $\left(11,12\right)$.

Find $\mathrm{AB}$.

Point $\mathrm{M}$ is the midpoint of $\left[\mathrm{AB}\right]$.

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

A circular pizza is divided into slices with an angle at the centre of 30°. Each slice has an area of $\frac{3\pi }{16}$ cm².

Find the exact perimeter of each slice.

The points A, B and C, located on the grid below, form a triangle.

Find BC.

Find the coordinates of the midpoint, M, of [BC].

Calculate the area of the triangle ABM.

On the diagram below, points $\mathrm{A}\left(-2,\frac{11}{2}\right)$ and $\mathrm{B}\left(\frac{1}{2},\frac{1}{2}\right)$ indicate the position of metal spikes called pitons that aid mountaineers as they climb. Each unit on the graph is 16 cm in real life.

A third piton is required halfway between points A and B at point M.

Calculate AM.

Pitons are to be placed along a mountain slope of length $25\sqrt{5}$ m, spaced at the distance AM found in part (a), with a piton at each end of the slope.

Find the number of pitons required.

The diagram below shows the sector of a circle $\mathrm{OAB}$.

(i) Find the area of the sector $\mathrm{OAB}$, giving your answer to 3 significant figures.

(ii) Find the area of the triangle $\mathrm{OAB}$, giving your answer to 3 significant figures.

(iii) Find the area of the shaded segment, giving your answer to 3 significant figures.

(i) Find the length of the arc $\mathrm{AB}$.

(ii) Find the perimeter of the sector $\mathrm{OAB}$.

The canopy of a parachute and the outermost connecting cords form a sector of a circle, $\mathrm{OAB}$, as shown in the following diagram, with the parachutist modelled as a particle at point $\mathrm{O}$.

The angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\frac{\pi }{4}$ radians and the area of sector $\mathrm{OAB}$ is $\frac{81\pi }{200}$ m².

Find the length of one of the connecting cords.

A plastic puzzle piece is in the form of a prism with a cross-section that is a sector of a circle, as shown in the following diagram. The radius of the sector is 8 cm and the angle at the centre is 1.2 radians.

The height of the puzzle piece is 2 cm.

(i) Find the area of the cross-section.

(ii) Hence find the volume of the puzzle piece.

A sector $\mathrm{OPQ}$ of a circle has $\mathrm{P}\hat{\mathrm{O}}\mathrm{Q}=\frac{5\pi }{6}$ radians.

The area of sector $\mathrm{OPQ}$, in cm², is one-fifth of the length of arc $\mathrm{PQ}$, in cm.

(i) Show that the radius of the sector is 0.4 cm.

(ii) Hence find the area of sector $\mathrm{OPQ}$ and the length of arc $\mathrm{PQ}$.

Give your answers to part (ii) correct to 3 significant figures.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$ and radius 9.8 cm, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=0.85$ radians.

Find the area of the shaded segment, bounded by arc $\mathrm{AB}$ and chord $\left[\mathrm{AB}\right]$. Give your answer correct to 3 significant figures.

Find the perimeter of sector $\mathrm{OAB}$.

A games design company produces a popular game called ‘Inconsequential Endeavour’. Each game set includes solid plastic game pieces in the form of a right prism with a cross-section that is a sector of a circle, as shown in the following diagram. The angle at the centre of the sector is 1.05 radians and the height of the game piece is 4 mm.

Given that the volume of the game piece is 0.412 cm³, find the radius of the sector. Give your answer correct to 3 significant figures.

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 $y$.

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 $\mathrm{R}$.

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 $\mathrm{A}\left(x,-5\right)$ 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 $y$.

A sector $\mathrm{OPQ}$ of a circle has $\mathrm{P}\hat{\mathrm{O}}\mathrm{Q}=\frac{11\pi }{14}$ radians.

The area of sector $\mathrm{OPQ}$, in cm², is equal to the length of arc $\mathrm{PQ}$, in mm.

Find

(i) the area of sector $\mathrm{OPQ}$, in cm²

(ii) the length of arc $\mathrm{PQ}$, in cm.

Give your answers correct to 3 significant figures.

The following diagram shows the sector $\mathrm{OAB}$ of a circle with centre $\mathrm{O}$, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=1.3$ radians. The shaded region is the segment bounded by arc $\mathrm{AB}$ and chord $\left[\mathrm{AB}\right]$.

Given that the difference between the perimeter of sector $\mathrm{OAB}$ and the perimeter of triangle $\mathrm{OAB}$ is 1.05 cm, find the area of the shaded region. Give your answer correct to 3 significant figures.

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.

A games design company produces a popular game called ‘Nugatory Enterprise’. Each game set includes plastic game pieces in the form of a right prism with a cross-section that is a sector of a circle, as shown in the following diagram. The angle at the centre of the sector is 0.9 radians and the height of the game piece is 6 mm.

The game pieces are hollow, with a top (which is a cross-section of the prism) and three sides, but no bottom.

Given that the external surface area of the game piece is 4.26 cm², find the interior volume of the game piece. Give your answer correct to 3 significant figures. You may ignore the thickness of the top and sides in your calculations.

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 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 $\mathrm{OAB}$.

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.