Angles and triangles 177
508
708
c 5 12.0 cm
B
a
C
A
508
608
f 5 5.0 cm
d 5 4.42 cm
E
F
D
Figure 20.39
In triangle ABC, 50 ◦ + 70 ◦ + ∠C = 180 ◦ , from which
∠C = 60 ◦ .
In triangle DEF, ∠E = 180 ◦ − 50 ◦ − 60 ◦ = 70 ◦ .
Hence, triangles ABC and DEF are similar, since their
angles are the same. Since corresponding sides are in
proportion to each other,
a
d
=
c
f
i.e.
a
4.42
=
12.0
5.0
Hence, side, a =
12.0
5.0
(4.42) = 10.61 cm.
Problem 27. In Figure 20.40, find the dimensions
marked r and p
558
358
x 5 7.44 cm
q 5 6.82 cm
z 5
1 2 . 9 7 c m
y 5 1 0 .6 3 c m
Y
Q
P
R
Z
X
p
r
Figure 20.40
In triangle PQR, ∠Q = 180 ◦ − 90 ◦ − 35 ◦ = 55 ◦ .
In triangle XYZ, ∠X = 180
◦
− 90
◦
− 55
◦
= 35
◦ .
Hence, triangles PQR and ZYX are similar since their
angles are the same. The triangles may he redrawn as
shown in Figure 20.41.
358
558
y 5 10.63 cm
q 5 6.82 cm
x
5 7.44 cm
z 5 1 2 . 9 7 c m
Z
X
Y
358
558
P
R
p
r
Q
Figure 20.41
By proportion:
p
z
=
r
x
=
q
y
i.e.
p
12.97
=
r
7.44
=
6.82
10.63
from which,
r = 7.44
6.82
10.63
= 4.77 cm
By proportion:
p
z
=
q
y
i.e.
p
12.97
=
6.82
10.63
Hence,
p = 12.97
6.82
10.63
= 8.32 cm
Problem 28. In Figure 20.42, show that triangles
CBD and CAE are similar and hence find the length
of CD and BD
B
10
6
9
12
C
D
E
A
Figure 20.42
Since BD is parallel to AE then ∠CBD = ∠CAE and
∠CDB = ∠CEA (corresponding angles between parallel lines). Also, ∠C is common to triangles CBD and
CAE.
Since the angles in triangle CBD are the same as in
triangle CAE, the triangles are similar. Hence,
by proportion:
CB
CA
=
CD
CE
=
BD
AE
i.e.
9
6 + 9
=
CD
12
, from which
CD = 12
9
15
= 7.2 cm
Also,
9
15
=
BD
10
, from which
BD = 10
9
15
= 6 cm