C11 - Lines & Planes In 3d

MATHEMATICS SPM LINES AND PLANES IN 3-DIMENSIONS

“Open and Close” Method to identify i. Angles between Lines and Planes ii. Angles between Two Planes

MOHD NAZAN BIN KAMARUL ZAMAN SMK. KOTA KLIAS BEAUFORT, SABAH

Use “Open and Close” Method to identify angles between lines and planes * Rule to remember – i) three points when joint together must be a straight line * A. Angles between Lines and Planes Example 1 The following diagram shows a cuboid KLMNPQRS S

R 4 cm

P

Q N

M 5 cm

K

12 cm

L

a) Identify the angle between the line LP and the plane KLMN b) Identify the angle between the line SL and the plane KLMN Solution (a) i. Draw the line LP and the plane KLMN P

N

M

L

K

ii. ‘Close’ the line LP and the plane KLMN together.

P

N

M

P K

L

The line PL lies straight to line KL, so the straight line we use is line KL

iii. ‘Open’ the line LP and the plane KLMN (180º) N

M 180º

K

P

L

Choose three point that when it joint together become a straight line That the three points are point K, point L and point P The straight line KLP or PLK, so the angle between the line LP and the plane KLMN is ∠ KLP or ∠ PLK Solution b i. Draw the line SL and the plane KLMN S

N

M

K

L

ii. ‘Close’ the line SL and the plane KLMN together. S

N K

M L

The line SL lies straight to point N and point L, so the straight line we use is line NL

iii. ‘Open’ the line SL and the plane KLMN(180º)

N K

M L S

Choose three point that when it joint together become a straight line That the three points are point N, point L and point S The straight line NLS or SLN, so the angle between the line SL and the plane KLMN is ∠ NLS or ∠ SLN

Use “Open and Close” Method to identify angles between two planes B. Angles between Two Planes Example 1 The following diagram shows a right prism ABCHED with a horizontal plane ABCD. It is a uniform prism and its cross section is an isosceles triangle of sides 4 cm. The thickness of the prism, EA = 4 cm. H

E

C

D

B

A

Identify the angle between the plane ABH and the plane ABCD.

Solution i. Draw the plane ABH and the plane ABCD H

C

D

A

B

ii. ‘Open’ the plane ABH (180º) H D D

C

B

A

C

A

B

H

H

Choose three point that when it joint together become a straight line Point D, point A and point H when joint together became a straight line but point C, point B and point H when joint together it is not a straight line, so we choose point D, point A and point H. H

E

C

D

A

B

Therefore the angle between the two planes is ∠ DAH @ ∠ HAD

Example 2 The diagram shows a right prism on a horizontal plane SRUT. Equilateral triangle RUQ and STP are the uniform cross-section of the prism. M and N are the mid-points of ST and RU, respectively. P

Q

T M

S

N

R

U

Name the angle between the plane PRN and the plane RSTU

Solution i. Draw the plane PRN and the plane RSTU P

S

T

•M R

•N

U

ii. ‘Open’ the line SL and the plane KLMN (180). P

S

T

•M

•N

R

S

•M

T

R

•N

U

U

P

P

Choose three point that when it joint together become a straight line Point M, point N and point P when joint together became a straight line Therefore the angle between the two planes is ∠ MNP @ ∠ PNM

TRY THIS A. Angles between Lines and Planes Identify the angle between the line BE and the plane ABCD Draw the line and the plane

1

H

F

E G

D A

C B

2

F

E

C

D

B

A E

3

B A

C D

‘Close’ the line and the plane ‘Open’ the line (180º) together

B. Angles between Two Planes Identify the angle between two shaded planes Draw shaded two plane

1

V

D

C

A

B A

2 B

D C

F G

E H

3

T

P

Q

S

R

‘Close’ the line and the plane ‘Open’ the line (180º) together

PAST YEAR QUESTIONS Nov 2003 1. Diagram 1 shows a prism with a horizontal square base HJKL. Trapezium EFLK is the uniform cross-section of the prism. The rectangular surface DEKJ is vertical while the rectangular surface GFLH is incline. 2. D G

F

E

H J

6 cm

K

L

8 cm

Diagram 1 Calculate the angle between the plane DLH and the base HJKL. July 2004, Q4

3. Diagram 2 shows a cuboid. E

H

D

C

9 cm

F

G

5 cm DIAGRAM 2 A

SMKKK

12 cm

B

[ 4 marks ]

Calculate the angle between the line AH and the plane ABCD.

[4 marks]

Nov 2004, Q3 4. Diagram 2 shows a pyramid VJKLM. V

L

M

Q cm

10 cm K

J

12 cm

DIAGRAM 2 The base JKLM is a horizontal rectangle. Q is the midpoint of JM. The apex V is 8 cm vertically above the point Q. Calculate the angle between the line KV and the base JKLM.

[ 4 marks ]

July 2005, Q2 5. Diagram 1 shows a right prism with rectangle ABCD as its horizontal base. Right angled triangle FAB is the uniform cross-section of the prism. The rectangular surface BCEF is inclined. E

F D

C

3 cm 12 cm A

SMKKK

5 cm

B

DIAGRAM 1

Calculate the angle between the plane ABE and the base ABCD. Nov 2005, Q4

[3 marks]

6. Diagram 1 shows a right prism. Right angled triangled PQR is the uniform cross-section of the prism. T

5 cm

12 cm

S

U DIAGRAM 1 18 cm Q

R

P Calculate the angle between the plane RTU and the plane PQTU. July 2006, Q4 7. Diagram 2 shows a right prism. The base HJKL is a horizontal rectangle. The right angled triangle NHJ is the uniform cross-section of the prism. M N DIAGRAM 2 8 cm

H

SMKKK

L

12 cm

K 6 cm J

Identify and calculate the angle between the line KN and the plane HLMN. Nov 2006, Q2 7. Diagram 1 shows a right prism. The base PQRS is on horizontal rectangle. The right triangle UPQ is the uniform cross section of the prism.

Identify and calculate the angle between the line RU and the base PQRS. [ 4 marks ] NOV 2007, Q4

Diagram 1 shows a right prism. The base PQRS is a horizontal rectangle. Right angled triangle QRU is the uniform cross-section of the prism. V is the midpoint of PS. T

S

U 5cm

V

R P

16cm 12cm Q

SMKKK

Identify and calculate the angle between the line UV and the plane RSTU

SMKKK

[3marks]