Bearings, Scale Drawing, Constructions & Loci Flashcards

All 28 cards in this deck

What is a bearing?

An angle measured from the north line, clockwise, written with three figures.
e.g. $072°$

What are the steps to measure a bearing from a diagram with a protractor?
e.g. measure the bearing of $B$ from $A$

1. Draw a north line at the point you measure from ($A$).
2. Put the protractor centre on $A$ with $0°$ on the north line.
3. Measure clockwise to the line $AB$ and write it with three figures, e.g. $072°$.

How do you find the back bearing (the bearing of $A$ from $B$) when you know the bearing of $B$ from $A$?
e.g. the bearing of $B$ from $A$ is $147°$

Add $180°$ if the bearing is less than $180°$, subtract $180°$ if it is $180°$ or more.
$147 + 180 = 327°$

The north lines drawn at two different points are parallel. What angle fact does this give you when working with bearings?

Co-interior (allied) angles between the two north lines and the line joining the points add up to $180°$.

True or false? The bearing of $A$ from $B$ is measured from a north line drawn at $B$.

True. You always measure from the north line at the point you are travelling from.

True or false? The bearing of $A$ from $B$ is always $180°$ more than the bearing of $B$ from $A$.

False. You add $180°$ only if the first bearing is under $180°$; otherwise you subtract $180°$, so the answer stays between $0°$ and $360°$.

What does a scale of $1:50$ mean?

$1$ unit on the drawing represents $50$ of the same units in real life.

How do you convert a length measured on a map into the real length?
e.g. $4$ cm on a map of scale $1:25\,000$

Multiply the measured length by the scale factor.
$4 \times 25\,000 = 100\,000$ cm $= 1$ km

How do you convert a real length into the length to draw on a scale drawing?
e.g. real length $40$ m, scale $1$ cm represents $5$ m

Divide the real length by the scale.
$40 \div 5 = 8$ cm

For a scale of $1:n$, what is the scale factor for areas?

$n^2$.
e.g. for $1:50$ the area scale factor is $2500$.

What are the steps to plot a point on a scale drawing from a distance and a bearing?
e.g. $B$ is $6$ km from $A$ on a bearing of $070°$, scale $1$ cm represents $2$ km

1. Convert the real distance to a drawing length: $6 \div 2 = 3$ cm.
2. Draw a north line at $A$ and measure $070°$ clockwise from it.
3. Draw $3$ cm along that direction and label the end $B$.

True or false? On a plan with scale $1:100$, an area of $1\text{ cm}^2$ on the plan represents $100\text{ cm}^2$ of real area.

False. Areas scale by $100^2 = 10\,000$, so it represents $10\,000\text{ cm}^2$.

What are the steps to construct a triangle given its three sides (SSS)?
e.g. sides $7$ cm, $5$ cm and $4$ cm

1. Draw the longest side as the base: $7$ cm.
2. Set compasses to $5$ cm and draw an arc from one end; set to $4$ cm and draw an arc from the other end.
3. Join where the arcs cross to both ends of the base.

What are the steps to construct a triangle given two sides and the angle between them (SAS)?
e.g. sides $6$ cm and $5$ cm with an angle of $50°$ between them

1. Draw the $6$ cm side as the base.
2. Use a protractor at one end to measure $50°$ and draw a faint ray.
3. Mark $5$ cm along the ray and join that point to the other end of the base.

What are the steps to construct a triangle given two angles and the side between them (ASA)?
e.g. a side of $8$ cm with angles $40°$ and $65°$ at its ends

1. Draw the $8$ cm side as the base.
2. Measure $40°$ at one end and draw a line; measure $65°$ at the other end and draw a line.
3. Where the two lines cross is the third vertex.

You are given all three sides of a triangle. Which instrument fixes the position of the third vertex?

Compasses — draw arcs with radii equal to the two remaining sides from the ends of the base.

True or false? You should rub out the construction arcs once a triangle construction is finished, to make it neat.

False. Leave all the construction arcs showing — they are the evidence of how the triangle was constructed.

What are the steps to construct the perpendicular bisector of a line segment?
e.g. the line segment $AB$

1. Open compasses to more than half of $AB$; draw arcs above and below from $A$.
2. Keeping the same radius, draw arcs from $B$ to cross them.
3. Join the two crossing points — this line cuts $AB$ in half at $90°$.

What are the steps to construct the bisector of an angle?
e.g. the bisector of angle $ABC$

1. With compasses centred on $B$, draw an arc crossing both arms.
2. From each crossing point, draw equal arcs that intersect.
3. Join $B$ to that intersection — it splits the angle in two.

What are the steps to construct the perpendicular from a point to a line, using ruler and compasses?
e.g. from point $P$ to the line $CD$

1. With compasses centred on $P$, draw an arc crossing $CD$ twice.
2. From each of those two points, draw equal arcs that meet on the other side of $CD$.
3. Join $P$ to that intersection.

What are the steps to construct an angle of $90°$ at a point on a line?
e.g. at the point $P$ on the line $AB$

1. With compasses centred on $P$, mark equal distances either side of $P$ on the line.
2. From each of those two marks, draw larger equal arcs that cross above the line.
3. Join $P$ to that crossing point.

What is the shortest distance from a point to a line?

The perpendicular distance from the point to the line.

What is a locus?

The set of all points that satisfy a given rule or condition.

What is the locus of points a fixed distance from a fixed point?

A circle with that radius, centred on the point.

What is the locus of points a fixed distance from a line segment?

Two parallel lines that distance either side, joined by a semicircle of that radius at each end (a 'racetrack' shape).

What is the locus of points equidistant from two points $A$ and $B$?

The perpendicular bisector of $AB$.

What is the locus of points inside an angle that are equidistant from its two arms (the two lines forming the angle)?

The bisector of the angle between the two lines.

What are the steps to shade a region satisfying two loci conditions on a scale diagram?
e.g. less than $4$ cm from $A$ and nearer to $B$ than to $C$

1. Construct each boundary: a circle of radius $4$ cm centred on $A$, and the perpendicular bisector of $BC$.
2. Decide which side of each boundary fits the condition (inside the circle; the $B$ side of the bisector).
3. Shade only the overlap of those regions.