triangle. ratio of the area of triangles theorem 1
TRANSCRIPT
Triangle
FE
D
CB
A
Ratio of the area of triangles
1l2l
1A2A
2
1
A
A 2
2
1 )(l
lTheorem 1
Example
In the figure, BC// DE, AC= 3 cm and CE= 4 cm.
FindADEofarea
ABCofarea
ADEofarea
ABCofarea
2
7
3
49
9
D
4 cm3 cmEC
B
A
2
AE
AC
Class work
In the figure, PSQ, QXR and RYP are straight lines.
If the area of is , find the area of the parallelogram SXRY.
PQR 2225 cm
Y
X
S
RQ
P
30 cm20 cm
Y
X
S
RQ
P
30 cm20 cm
Identify the similar triangles first.PQRPSY ~
2
QR
SY
PQRofarea
PSYofarea
2
50
30
225
PSYofarea
81PSYofarea
81
Y
X
S
RQ
P
30 cm20 cm
81
PQRSQX ~
2
QR
QX
PQRofarea
SQXofarea
2
50
20
225
SQXofarea
36SQXofarea 36
Area of parallelogram = 225 – 81 - 36
= 1082108 cmArea of parallelogram is
TriangleC
BA
Triangle
baseA B
C
height
C
BA
Triangle
C
BA
base
height
Triangle
C
BA
Triangle
C
BA
Triangle
baseheight
Area of Triangle
baseA B
C
height
heightbase2
1Area
D CB
A
What is the relationship between the heights of
and ?ABD ADC
h
D CB
A
Triangles have common height h
ADCofarea
ABDofarea
h
D CB
A
For triangles with common height,
find
hDC
hBD
2
12
1
DC
BD
For triangles with common height,
D CB
A
ADCofarea
ABDofarea
2
1
b
bTheorem 2
D CB
A
Eg.3) Given that BC : DC = 5: 1
ADCofarea
ABDofarea
Find
D CB
A
Eg.3) Given that BC : DC = 5: 1
ADCofarea
ABDofarea
Find
ABD ADCand have common height,
ADCofarea
ABDofarea
DC
BD
D CB
A
Eg.3) Given that BC : DC = 5: 1
ADCofarea
ABDofarea
Find
ABD ADCand have common height,
ADCofarea
ABDofarea
DC
BD
1
4
= 4
D1 cm
3 cm
CB
A
Eg.4) Given that AD = 3 cm and CD = 1 cm.
Find BDCofarea
ABDofarea
D1 cm
3 cm
CB
A
Eg.4) Given that AD = 3 cm and CD = 1 cm.
Find BDCofarea
ABDofarea
ABD ADCand have common height,
BDCofarea
ABDofarea
CD
AD
1
3
3
Class work5.) In the figure, find the area of : area of .PRX QRX
xx
4 cm5 cm
X
R
QP
Class work5.) In the figure, find the area of : area of .PRX QRX
xx
4 cm5 cm
X
R
QP
heightcommonthehaveRXQandPRX
Class work5.) In the figure, find the area of : area of .PRX QRX
RXQofarea
PRXofarea
x
x
= 1xx
4 cm5 cm
X
R
QP
6.) In the figure, given that QX:XR = 5:6 and PY:YR =5:3
calculate the area of (a) (b)
PXR
RXY
Y
X RQ
P
244 cmisPQRofareatheif
6.) In the figure, given that QX:XR = 5:6 and PY:YR =5:3
Y
X RQ
P
244 cmisPQRofareatheif
5x 6x
heightcommonthehavePQRandPQXPXR ,
PQRofarea
PXRofarea
xx
x
65
6
x
x
11
6
11
6
6.) In the figure, given that QX:XR = 5:6 and PY:YR =5:3
Y
X RQ
P
244 cmisPQRofareatheif
5x 6x
heightcommonthehavePQRandPQXPXR ,
PXRofarea 4411
6 24
6.) In the figure, given that QX:XR = 5:6 and PY:YR =5:3
Y
X RQ
P
244 cmisPQRofareatheif
5x 6x
heightcommonthehaveRXYandPXR
PXRofarea
RXYofarea
yx
y
35
3
y
y
8
3
8
3
5y
3y
6.) In the figure, given that QX:XR = 5:6 and PY:YR =5:3
Y
X RQ
P
244 cmisPQRofareatheif
5x 6x
RXYofarea 248
3 9
5y
3y
N
M
S
RQ
P
7) In the figure, PQRS is a rectangle.
M is a midpoint of QR. PR and MS intersects at N.
Find the area of : area of PQMN.NRS
In the figure, PQRS is a rectangle.
N
M
S
RQ
P
Find the area of : area of PQMN.
M is a midpoint of QR. PR and MS intersects at N.
NRS
M is a midpoint of QR. PR and MS intersects at N.
In the figure, PQRS is a rectangle.
xx
N
M
S
RQ
PFind the area of : area of PQMN.NRS
2x
xx
N
M
S
RQ
P
Find the area of : area of PQMN.NRS
RNMPNS ~2
RM
PS
RNMofarea
PNSofarea2
2
x
x= 4
ARNMofareaLet APNSofareaThen 4,
A
4A
Find the area of : area of PQMN.NRS
4A
Ay
2y
Considering RNSandMNR They have the common height.
RNSofarea
MNRofarea
AMNRofareaLet ARNSofareathen 2
2A
y
y
2
2
1
Find the area of : area of PQMN.NRS
4A
Ay
2y
APSRofarea 6
AA 6
2A
PQRofarea
Hence, A5
= 2A : 5A = 2 : 5
PQMNofareaNRSofarea :
,PRSgConsiderin ,RSPPQRSince
PQMNofarea
A6
Y
X
R
Q
P
8.) In the figure, PX:XQ = 1: 2, PY:YR = 3:2.Area of : Area of = ?QXY PQR
In the figure, PX:XQ = 1: 2, PY:YR = 3:2.Area of : Area of = ?QXY PQR
PXY XYQand have the common height
2x
x
Y
X
R
Q
P
XYQofarea
PXYofarea
2
1
In the figure, PX:XQ = 1: 2, PY:YR = 3:2.Area of : Area of = ?QXY PQR
PXY XYQand have the common height
APXYofarealet 2A
A
2x
x
Y
X
R
Q
P
XYQofareathen
2
1
XYQofarea
PXYofarea
A2
In the figure, PX:XQ = 1: 2, PY:YR = 3:2.Area of : Area of = ?QXY PQR
PYQ QYRand have the common height
APYQofarea 3
QYRofareathen
QYRofarea
PYQofarea
2y3y
2A
A
2x
x
Y
X
R
Q
P
= 2A : 5A
= 2 : 5
2
3
A2
PQRofareaQXYofArea :
Y
X
S
RQ
P
9.) In the figure, PQRS is a rectangle. RSX is a straight line and PX// QS. If the area of PQRS is 24 and Y is a point on QR such that QY :YR = 3:1, find the area of . SXY
X
S R
QP
3
1
QRXofarea
RSXofarea10.) In the figure, if ,
Find PQRStrapeziumofarea
RSXofarea
X
S R
QP
3
1
QRXofarea
RSXofarea10.) In the figure, if ,
heightcommonthehaveQRXandRSX
A
3A
X
S R
QP
3
1
QRXofarea
RSXofarea10.) In the figure, if ,
3
1
XQ
SX
x
3x
x
x
3
A
3A
X
S R
QP
3
1
QRXofarea
RSXofarea10.) In the figure, if ,
x
3x
,similararePQXandRSXSince
PQXofarea
RSXofarea
2
QX
SX2
3
x
x
9
1
A
3A
9A
X
S R
QP
3
1
QRXofarea
RSXofarea10.) In the figure, if ,
x
3x
,heightandbasesamethehaveRPQandSPQSince
A
3A
9A
areaequalhaveRPQandSPQ
SPQofarea A3
3A
X
S R
QP
3
1
QRXofarea
RSXofarea10.) In the figure, if ,
x
3x
A
3A
9A
3A
PQRStrapeziumofarea
RSXofarea AAAA
A
393
A
A
16
16
1
FE
D
CB
A
Ratio of the area of
similar triangles
1l2l
1A2A
2
1
A
A 2
2
1 )(l
l
Theorem 1
For triangles with common height,
D CB
A
ADCofarea
ABDofarea
2
1
b
b
Theorem 2