Loci Exercises 201y1j
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Loci Exercises 1. Mr Dumpleton is 2cm from shape Q. Shade the region he could be in.
Q
2. Sketch the region in which you are at most 2cm from shape A.
www.drfrostmaths.com
3.
A
4. Draw the locus representing points which are 1cm from the edges of polygon M (this could include the
inside).
7. Find the locus for which the points are equidistant from lines A and B.
A
M
A
5. Sketch the region which is at most 5cm from A and 3cm from B.
B
B
8. Draw the locus representing points which are equidistant from A and B.
6.
A
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B
A
B
10.Sketch the region at most 3cm away from A and at most 2cm away from B. 9. Mr Belemet is tied by a rope, of length 4cm to a fixed point A. Shade the region in which Mr Belemet can graze.
A B A
11.Sketch the region where you are at most 2.5cm from A, at least 2cm from B, and at most 1.5cm from C.
A C
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B
12.
c. Closer to AB than to AD, less than 4cm away from A, and more than 1cm away from CD.
13. 14.
A
B
D
C
15.Shade the region within rectangle ABCD which is: a. Closer to AB than to CD, and closer to BC than to AB.
A
B
d. Closer to BC than to AD, more than 3cm away from B, and closer to AB than to BC.
D
C
A
B
D
C
b. Closer to AB than to CD, and at most 3cm away from A.
A
B
16. For the following questions, calculate the area of the locus, in of the given variables (and
D
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C
π
where appropriate). Assume that you could be inside or outside the shape unless otherwise specified.
e.
x
j.
metres away from the edges of a square
of length
Being attached to one corner on the outside of
x×x
l .
square building (which you can’t go
2x .
inside), by a rope of length ____________________ f.
x
metres away from the edges of a
w
rectangle of sides
and
____________________
h .
k. At most
x
metres away from an L-shaped
building with two longer of longer sides ____________________ g.
x
and four shorter sides of
metres away from the edges of an
equilateral triangle of side length
y .
h. Inside a square ABCD of side being at least
x
Being attached to one corner on the outside of
w×h
x
metres,
square building (which you can’t go
inside), by a rope of length
metres from A, and closer
between the cases when ____________________ Being inside an equilateral triangle of side
2 x , and at least vertices.
____________________
www.drfrostmaths.com
x
away from each of the
x
(where
x< w+h ). You may wish to distinguish
to BC than to CD.
i.
metres.
____________________ l.
____________________
x
2x
x< h 17.
and otherwise.
x< w
and/or
18.
19.
Answers
1. Mr Dumpleton is 2cm from shape Q. Shade the region he could be in.
Q
2. Sketch the region in which you are at most 2cm from shape A.
www.drfrostmaths.com
A
3. Draw the locus representing points which are 1cm from the edges of polygon M (this could include the
inside).
www.drfrostmaths.com
4. Sketch the region which is at most 5cm from A and 3cm from B. Two circles of radius 5cm and 3cm drawn. Overlap shaded. 20. 5. Find the locus for which the points are equidistant from lines A and B. Angle bisectors with appropriate construction lines. 21.
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6. Draw the locus representing points which are equidistant from A and B. 7. Mr Belemet is tied by a rope, of length 4cm to a fixed point A. Shade the region in which Mr Belemet can graze.
A
B
A
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8. Sketch the region at most 3cm away from A and at most 2cm away from B.
A B
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M
9. Sketch the region where you are at most 2.5cm from A, at least 2cm from B, and at most 1.5cm from C.
A
b. Closer to AB than to CD, and at most 3cm away from A.
A
B
D
C
B
C
c. Closer to AB than to AD, less than 4cm away from A, and more than 1cm away from CD. 22.Shade the region within rectangle ABCD which is: a. Closer to AB than to CD, and closer to BC than to AB.
A
B
D
C
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B
A
where appropriate). Assume that you could be inside or outside the shape unless otherwise specified. a.
x
metres away from the edges of a square
of length
l .
4 exterior rectangles:
D
4 xl
4 quarter circles forming 1 full circle:
C
π x2 4 interior rectangles:
4 xl
Total overlap on interior rectangles:
4 x2 d. Closer to BC than to AD, more than 3cm away from B, and closer to AB than to BC.
Total: b.
A
B
x
8 xl+π x 2−4 x 2
metres away from the edges of a
rectangle of sides
w
and
h .
Using the same approach as above, Area: c.
D
C
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metres away from the edges of an
equilateral triangle of side length
23. For the following questions, calculate the area of the locus, in of the given variables (and
x
4 xw +4 xl+ π x 2−4 x2
π
3 exterior rectangles:
y .
3 xy
3 sixth circles which form a semicircle:
1 2 πx 2
1 2 πx 2
3 interior rectangles (without overlap): Total:
3 x ( y−2 √ 3 x ) 6 interior corner right-angled triangles:
3 √3 x
2
Total:
f.
Being attached to one corner on the outside of
x×x 1 π x 2+ 6 xy−3 √ 3 x 2 2
x
square building (which you can’t go
inside), by a rope of length
d. Inside a square ABCD of side being at least
1 2 x ( 2 √ 3−π ) 2
x
metres,
metres from A, and closer
to BC than to CD. First calculate area of square minus area of quarter circle:
1 2 2 x − π x Half it: 4
3 4
2x .
of a circle with radius
2x :
Two quarter circles of radius a semicircle:
Total:
x
3 π x2 forming
1 π x2 2
7 π x2 2
1 2 x ( 8−π ) 8 e. Being inside an equilateral triangle of side
2 x , and at least
x
away from each of the
vertices. Area of entire triangle:
√3 x
2
Area of 3 sixth-circles forming semicircle:
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g. At most
x
metres away from an L-shaped
building with two longer of longer sides and four shorter sides of
x
metres.
2x
Five quarter-circles of radius
x :
3 2 πx 4
:
5 π x2 4
x> w , then we have an additional
If
Two rectangles:
4x
Three squares:
3 x2
2
quarter circle with area Similarly, if
1 2 π ( x−w ) . 4
x> h , we have an additional
2
Total:
1 ( x 5 π + 28 ) 4
quarter circle with area
h. Being attached to one corner on the outside of
w×h
If we let
square building (which you can’t go
inside), by a rope of length
x
a
between the cases when
x< h
b , then the total is:
3 1 1 π x 2 +max π ( x−w )2 , 0 +max π ( x−h )2 , 0 4 4 4
(
)
(
)
and/or f
and otherwise.
Three quarters of a circle with radius
www.drfrostmaths.com
give the maximum of
(where
x< w+h ). You may wish to distinguish x< w
and
max ( a , b )
1 π ( x−h )2 4
x> w+h , then things start to get very
hairy!
x 24.
I
Q
2. Sketch the region in which you are at most 2cm from shape A.
www.drfrostmaths.com
3.
A
4. Draw the locus representing points which are 1cm from the edges of polygon M (this could include the
inside).
7. Find the locus for which the points are equidistant from lines A and B.
A
M
A
5. Sketch the region which is at most 5cm from A and 3cm from B.
B
B
8. Draw the locus representing points which are equidistant from A and B.
6.
A
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B
A
B
10.Sketch the region at most 3cm away from A and at most 2cm away from B. 9. Mr Belemet is tied by a rope, of length 4cm to a fixed point A. Shade the region in which Mr Belemet can graze.
A B A
11.Sketch the region where you are at most 2.5cm from A, at least 2cm from B, and at most 1.5cm from C.
A C
www.drfrostmaths.com
B
12.
c. Closer to AB than to AD, less than 4cm away from A, and more than 1cm away from CD.
13. 14.
A
B
D
C
15.Shade the region within rectangle ABCD which is: a. Closer to AB than to CD, and closer to BC than to AB.
A
B
d. Closer to BC than to AD, more than 3cm away from B, and closer to AB than to BC.
D
C
A
B
D
C
b. Closer to AB than to CD, and at most 3cm away from A.
A
B
16. For the following questions, calculate the area of the locus, in of the given variables (and
D
www.drfrostmaths.com
C
π
where appropriate). Assume that you could be inside or outside the shape unless otherwise specified.
e.
x
j.
metres away from the edges of a square
of length
Being attached to one corner on the outside of
x×x
l .
square building (which you can’t go
2x .
inside), by a rope of length ____________________ f.
x
metres away from the edges of a
w
rectangle of sides
and
____________________
h .
k. At most
x
metres away from an L-shaped
building with two longer of longer sides ____________________ g.
x
and four shorter sides of
metres away from the edges of an
equilateral triangle of side length
y .
h. Inside a square ABCD of side being at least
x
Being attached to one corner on the outside of
w×h
x
metres,
square building (which you can’t go
inside), by a rope of length
metres from A, and closer
between the cases when ____________________ Being inside an equilateral triangle of side
2 x , and at least vertices.
____________________
www.drfrostmaths.com
x
away from each of the
x
(where
x< w+h ). You may wish to distinguish
to BC than to CD.
i.
metres.
____________________ l.
____________________
x
2x
x< h 17.
and otherwise.
x< w
and/or
18.
19.
Answers
1. Mr Dumpleton is 2cm from shape Q. Shade the region he could be in.
Q
2. Sketch the region in which you are at most 2cm from shape A.
www.drfrostmaths.com
A
3. Draw the locus representing points which are 1cm from the edges of polygon M (this could include the
inside).
www.drfrostmaths.com
4. Sketch the region which is at most 5cm from A and 3cm from B. Two circles of radius 5cm and 3cm drawn. Overlap shaded. 20. 5. Find the locus for which the points are equidistant from lines A and B. Angle bisectors with appropriate construction lines. 21.
www.drfrostmaths.com
6. Draw the locus representing points which are equidistant from A and B. 7. Mr Belemet is tied by a rope, of length 4cm to a fixed point A. Shade the region in which Mr Belemet can graze.
A
B
A
www.drfrostmaths.com
8. Sketch the region at most 3cm away from A and at most 2cm away from B.
A B
www.drfrostmaths.com
M
9. Sketch the region where you are at most 2.5cm from A, at least 2cm from B, and at most 1.5cm from C.
A
b. Closer to AB than to CD, and at most 3cm away from A.
A
B
D
C
B
C
c. Closer to AB than to AD, less than 4cm away from A, and more than 1cm away from CD. 22.Shade the region within rectangle ABCD which is: a. Closer to AB than to CD, and closer to BC than to AB.
A
B
D
C
www.drfrostmaths.com
B
A
where appropriate). Assume that you could be inside or outside the shape unless otherwise specified. a.
x
metres away from the edges of a square
of length
l .
4 exterior rectangles:
D
4 xl
4 quarter circles forming 1 full circle:
C
π x2 4 interior rectangles:
4 xl
Total overlap on interior rectangles:
4 x2 d. Closer to BC than to AD, more than 3cm away from B, and closer to AB than to BC.
Total: b.
A
B
x
8 xl+π x 2−4 x 2
metres away from the edges of a
rectangle of sides
w
and
h .
Using the same approach as above, Area: c.
D
C
www.drfrostmaths.com
metres away from the edges of an
equilateral triangle of side length
23. For the following questions, calculate the area of the locus, in of the given variables (and
x
4 xw +4 xl+ π x 2−4 x2
π
3 exterior rectangles:
y .
3 xy
3 sixth circles which form a semicircle:
1 2 πx 2
1 2 πx 2
3 interior rectangles (without overlap): Total:
3 x ( y−2 √ 3 x ) 6 interior corner right-angled triangles:
3 √3 x
2
Total:
f.
Being attached to one corner on the outside of
x×x 1 π x 2+ 6 xy−3 √ 3 x 2 2
x
square building (which you can’t go
inside), by a rope of length
d. Inside a square ABCD of side being at least
1 2 x ( 2 √ 3−π ) 2
x
metres,
metres from A, and closer
to BC than to CD. First calculate area of square minus area of quarter circle:
1 2 2 x − π x Half it: 4
3 4
2x .
of a circle with radius
2x :
Two quarter circles of radius a semicircle:
Total:
x
3 π x2 forming
1 π x2 2
7 π x2 2
1 2 x ( 8−π ) 8 e. Being inside an equilateral triangle of side
2 x , and at least
x
away from each of the
vertices. Area of entire triangle:
√3 x
2
Area of 3 sixth-circles forming semicircle:
www.drfrostmaths.com
g. At most
x
metres away from an L-shaped
building with two longer of longer sides and four shorter sides of
x
metres.
2x
Five quarter-circles of radius
x :
3 2 πx 4
:
5 π x2 4
x> w , then we have an additional
If
Two rectangles:
4x
Three squares:
3 x2
2
quarter circle with area Similarly, if
1 2 π ( x−w ) . 4
x> h , we have an additional
2
Total:
1 ( x 5 π + 28 ) 4
quarter circle with area
h. Being attached to one corner on the outside of
w×h
If we let
square building (which you can’t go
inside), by a rope of length
x
a
between the cases when
x< h
b , then the total is:
3 1 1 π x 2 +max π ( x−w )2 , 0 +max π ( x−h )2 , 0 4 4 4
(
)
(
)
and/or f
and otherwise.
Three quarters of a circle with radius
www.drfrostmaths.com
give the maximum of
(where
x< w+h ). You may wish to distinguish x< w
and
max ( a , b )
1 π ( x−h )2 4
x> w+h , then things start to get very
hairy!
x 24.
I
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