loci exercises

8/19/2019 Loci Exercises

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QA

Loci Exercises

1. Mr Dumpleton is 2cm from shape Q. Shade the

region he could be in.

2. Sketch the region in which you are at most 2cm from

shape A.

3.

. Draw the locus representing points which are 1cm

from the edges of polygon M !this could include the

www.drfrostmaths.com


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M

A

"

A "

A

"

A

"

inside#.

$. Sketch the region which is at most $cm from A and

3cm from ".

%.

&. 'ind the locus for which the points are e(uidistant

from lines A and ".

). Draw the locus representing points which aree(uidistant from A and ".



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A

A

B

A "

*

+. Mr "elemet is tied by a rope, of length cm to a -ed

point A. Shade the region in which Mr "elemet can

gra/e.

10.Sketch the region at most 3cm away from A and at

most 2cm away from ".

11.Sketch the region where you are at most 2.$cm from

A, at least 2cm from ", and at most 1.$cm from *.



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A "

*D

A "

*D

A "

*D

A "

*D

12.

13.

1.

1$.Shade the region within rectangle A"*D which is

a. *loser to A" than to *D, and closer to "* thanto A".

b. *loser to A" than to *D, and at most 3cm

away from A.

c. *loser to A" than to AD, less than cm away

from A, and more than 1cm away from *D.

d. *loser to "* than to AD, more than 3cm away

from ", and closer to A" than to "*.

1%.

'or the following (uestions, calculate the area of

the locus, in terms of the gien ariables !and π

where appropriate#. Assume that you could be inside

or outside the shape unless otherwise speci-ed.



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e. x metres away from the edges of a s(uare

of length l .

f. x metres away from the edges of a

rectangle of sides w and h .

g. x metres away from the edges of an

e(uilateral triangle of side length y .

h. 4nside a s(uare A"*D of side x metres,

being at least x metres from A, and closer

to "* than to *D.

i. "eing inside an e(uilateral triangle of side

2 x

, and at least

x

away from each of theertices.

5. "eing attached to one corner on the outside of

x × x s(uare building !which you can6t go

inside#, by a rope of length 2 x .

k. At most x metres away from an 78shaped

building with two longer of longer sides 2 x

and four shorter sides of x metres.

l. "eing attached to one corner on the outside of

w × h s(uare building !which you can6t go

inside#, by a rope of length x !where

x


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QA

18. Answers

1. Mr Dumpleton is 2cm from shape Q. Shade the

region he could be in.

2. Sketch the region in which you are at most 2cm from

shape A.

1+.

3. Draw the locus representing points which are 1cm

from the edges of polygon M !this could include the



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inside#.



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. Sketch the region which is at most $cm from A and

3cm from ".

Two circles of radius 5cm and 3cm drawn.

Overlap shaded.

20.

5. 'ind the locus for which the points are e(uidistantfrom lines A and ".

Angle isec!ors wi!h appropria!e cons!ruc!ion

lines.

"1.



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A"

A

%. Draw the locus representing points which are

e(uidistant from A and ".&. Mr "elemet is tied by a rope, of length cm to a -ed

point A. Shade the region in which Mr "elemet can

gra/e.



10/14

M

A

B

). Sketch the region at most 3cm away from A and at

most 2cm away from ".



11/14

A "

*

A "

*D

A "

*D

+. Sketch the region where you are at most 2.$cm from

A, at least 2cm from ", and at most 1.$cm from *.

22.Shade the region within rectangle A"*D which isa. *loser to A" than to *D, and closer to "* than

to A".

b. *loser to A" than to *D, and at most 3cm

away from A.

c. *loser to A" than to AD, less than cm away

from A, and more than 1cm away from *D.



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A "

*D

A "

*D

d. *loser to "* than to AD, more than 3cm away

from ", and closer to A" than to "*.

23. 'or the following (uestions, calculate the area of

the locus, in terms of the gien ariables !and π

where appropriate#. Assume that you could be inside

or outside the shape unless otherwise speci-ed.

a. x metres away from the edges of a s(uare

of length l .

# ex!erior rec!angles$ 4 xl

# %uar!er circles forming 1 full circle$

π x2

# in!erior rec!angles$ 4 xl

To!al overlap on in!erior rec!angles$

4 x2

To!al$ 8 xl+π x2−4 x

2

b. x metres away from the edges of a

rectangle of sides w and h .

&sing !he same approach as aove'

Area$ 4 xw+4 xl+π x2−4 x

2

c. x metres away from the edges of an

e(uilateral triangle of side length y .

3 ex!erior rec!angles$ 3 xy

3 six!h circles which form a semicircle$



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1

2π x

2

3 in!erior rec!angles (wi!hou! overlap)$

3 x ( y−2√ 3 x )

* in!erior corner righ!+angled !riangles$

3√ 3 x2

To!al$1

2π x

2+6 xy−3√ 3 x

2

d. 4nside a s(uare A"*D of side x metres,

being at least x metres from A, and closer

to "* than to *D.

,irs! calcula!e area of s%uare minus area

of %uar!er circle$

x2−1

4π x

2

-alf i!$

1

8 x

2 (8−π )

e. "eing inside an e(uilateral triangle of side

2 x, and at least

x away from each of the

ertices.

Area of en!ire !riangle$ √ 3 x2

Area of 3 six!h+circles forming semicircle$

1

2π x

2

To!al$1

2 x

2 (2√ 3−π )

f. "eing attached to one corner on the outside of x × x s(uare building !which you can6t go

inside#, by a rope of length 2 x .

3

4 of a circle wi!h radius2 x $ 3π x

2

Two %uar!er circles of radius x forming

a semicircle$ 12π x 2

To!al$7

2π x

2

g. At most

x

metres away from an 78shaped

building with two longer of longer sides 2 x

and four shorter sides of x metres.



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,ive %uar!er+circles of radius x $

5

4π x

2

Two rec!angles$ 4 x2

Three s%uares$ 3 x2

To!al$1

4 x

2

(5π +28)

h. "eing attached to one corner on the outside of

w × h s(uare building !which you can6t go

inside#, by a rope of length x !where

xh ' we have an addi!ional

%uar!er circle wi!h area1

4π ( x−h )2

f we le! max (a , b ) give !he maximum of

a and b ' !hen !he !o!al is$

3

4π x

2+max( 14 π ( x−w )2,0)+max( 14 π ( x−h )2 ,0)

f x>w+h ' !hen !hings s!ar! !o ge! ver0

hair0

2.


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