Flashcards · 13 cards

Construction & loci

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Rule
How do you bisect a line $AB$ with ruler and compasses?
Answer
With the compasses open more than half of $AB$ , draw arcs from $A$ and from $B$ above and below the line. Join the two crossings.
Bisect a lineEqual arcs from A and B; join the crossings
From the lesson: Ruler-and-compasses constructions
Rule
How do you bisect an angle?
Answer
An arc from the vertex cuts both arms. From those two points, draw equal arcs that cross. Join the vertex to the crossing.
Bisect an angleArc from O, then equal arcs from C and D
From the lesson: Ruler-and-compasses constructions
Rule
How do you construct $60^\circ$ ?
Answer
An arc from the point cuts the line. From that crossing, an arc with the same radius cuts the first arc. The two arcs make an equilateral triangle.
Construct 60°Two arcs of the same radius make an equilateral triangle
From the lesson: Ruler-and-compasses constructions
Know it
How do you construct $30^\circ$ , $45^\circ$ , $90^\circ$ and $120^\circ$ ?
Answer
$30^\circ$ : bisect $60^\circ$ . $90^\circ$ : a perpendicular. $45^\circ$ : bisect $90^\circ$ . $120^\circ$ : two $60^\circ$ angles side by side.
From the lesson: Ruler-and-compasses constructions
Rule
How do you drop a perpendicular from a point $P$ to a line?
Answer
An arc from $P$ cuts the line at $Q$ and $R$ . Equal arcs from $Q$ and $R$ meet at $X$ . Join $PX$ .
Drop a perpendicularArc from P cuts the line; equal arcs from Q and R meet at X
From the lesson: Ruler-and-compasses constructions
Rule
How do you divide a line in the ratio 3 : 2?
Answer
Draw a line from one end and mark 5 equal steps along it. Join the last step to the other end, then draw a parallel to it through step 3.
Divide a line in a ratio5 equal steps and a parallel line divide BC in the ratio 3 : 2
From the lesson: Ruler-and-compasses constructions
Know it
Drawing a pair of arcs: what must stay the same?
Answer
The radius. Keep the compasses open the same amount for both arcs of a pair.
From the lesson: Ruler-and-compasses constructions
Rule
The locus of points a distance $d$ from a point $O$ ?
Answer
A circle, centre $O$ , radius $d$ .
From a pointA circle, centre O, radius d
From the lesson: Loci
Rule
The locus of points a distance $d$ from a line?
Answer
Two lines parallel to it, one on each side, each $d$ away.
From a lineTwo parallel lines, each d away
From the lesson: Loci
Rule
The locus of points equidistant from two points $A$ and $B$ ?
Answer
The perpendicular bisector of $AB$ .
Equidistant from A and BThe perpendicular bisector of AB: PA = PB
From the lesson: Loci
Rule
The locus of points equidistant from two crossing lines?
Answer
The bisectors of the angles between them.
Equidistant from two linesThe two bisectors of the angles between them
From the lesson: Loci
Rule
The locus of points $P$ with $\angle XPY = 90^\circ$ ?
Answer
The circle with $XY$ as its diameter.
A right angle on XYThe circle with XY as diameter
From the lesson: Loci
Which method?
WAEC 2012 · Paper 2 · Q9
Three towns $X$ , $Y$ and $Z$ are such that $Y$ is 20 km from $X$ and 22 km from $Z$ . Town $X$ is 18 km from $Z$ . A Health Centre is to be built to serve the three towns, located such that patients from $X$ and $Y$ always travel equal distances to it, while patients from $Z$ travel exactly 10 km. Using a scale of 1 cm to 2 km, find by construction, using a pair of compasses and ruler only, the possible positions of the Health Centre.
(i) In how many possible locations can the Health Centre be built? (ii) Measure and record the distances of the locations from town $X$ . (iii) Which of these locations would be convenient for all the three towns?
Which two loci do you draw?
Answer
Equal distances from $X$ and $Y$ : the perpendicular bisector of $XY$ . A fixed distance from $Z$ : a circle centred at $Z$ . The centre can go where they meet.
From the lesson: Loci Try the question