. . .
. .
. . .
.
. .
. . AE
BC DE
AC AE y
y y
2 4 1 9 4 3
3 7 2 4 2 4 3 7 4 3
2 4 3 7 4 3 6 6
4 3
#
#
= +
=
=
=
=
=
=
2. Prove DXYZ <;DWVZ.
Solution
( )
ZV XZ
ZW YZ
ZV XZ
ZW YZ
XZY WZV
35 15
7 3
14 6
7 3
vertically opposite angles
`
+ +
= =
= =
=
=
` since two pairs of sides are in proportion and their included angles are equal the triangles are similar
Ratio of intercepts
The following result comes from similar triangles.
When two (or more) transversals cut a series of parallel lines, the ratios of their intercepts are equal.
: :
AB BC DE EF BC
AB EF DE That is,
or
=
=
ch4.indd 166 7/9/09 12:35:06 AM
Proof
Then opposite sides of a parallelogramAlso (similarly)
ratios of intercepts on parallel lines
#
2. Evaluate x and y , to 1 decimal place.
Solution
Use either similar triangles or ratios of intercepts to fi nd x . You must use similar triangles to fi nd y .
. .
.
.
. .
.
. .
. .
.
. .
. x
x
y
y 5 8 3 4
2 7
3 4 2 7 5 8 4 6 7 1 3 4
2 7 3 4
3 4 6 1 7 1 12 7
#
#
=
=
=
= +
=
=
1. Find the value of all pronumerals, to 1 decimal place where
appropriate.
(a)
(b)
(c)
(d)
(e)
4.5 Exercise s
These ratios come from intercepts on parallel lines.
These ratios come from similar triangles.
Why?
ch4.indd 168 7/9/09 12:35:09 AM
(f)
14.3 a
46
c
19
c
115
c
46
c
x
c
9.1
25.7
8.9 y
(g)
2. Evaluate a and b to 2 decimal places.
3. Show that DABC and DCDE are similar.
4. EF bisects +GFD. Show that DDEF and DFGE are similar.
5. Show that DABC and DDEF are similar. Hence fi nd the value of y .
4.2 4.9
6.86 1.3
1.82 5.88
A
C D B
E F
yc
87c
52c
6. The diagram shows two
concentric circles with centre O . Prove that
(a) DOAB <;DOCD. If radius
(b) OC=5 9 cm. and radius OB=8 3 cm,. and the length of CD=3 7 cm,. fi nd the length of AB , correct to 2 decimal places.
7. (a) Prove that DABC <;DADE. Find the values of
(b) x and y ,
correct to 2 decimal places.
8. ABCD is a parallelogram, with CD produced to E . Prove that
CEB. D
; ABF <
D
ch4.indd 169 7/9/09 5:10:06 PM
9. Show that DAED <; DABC. Find the value of m .
10. Prove that DABC and DACD are similar. Hence evaluate x and y .
11. Find the values of all
pronumerals, to 1 decimal place.
(a)
(b)
(c)
(d)
(e)
12. Show that (a) BCAB
FGAF
=
(b) ACAB AGAF
=
(c) BDCE EGDF
=
13. Evaluate a and b correct to 1 decimal place.
14. Find the value of y to 2 signifi cant fi gures.
15. Evaluate x and y correct to 2 decimal places.
ch4.indd 170 7/25/09 3:04:58 AM
Pythagoras’ Theorem
DID YOU KNOW?
The triangle with sides in the proportion 3:4:5 was known to be right angled as far back as ancient Egyptian times. Egyptian surveyors used to measure right angles by stretching out a rope with knots tied in it at regular intervals.
They used the rope for forming right angles while building and dividing fi elds into rectangular plots.
It was Pythagoras (572–495 BC) who actually discovered the
relationship between the sides of the right-angled triangle. He was able to
generalise the rule to all right-angled triangles.
Pythagoras was a Greek mathematician, philosopher and mystic. He founded the Pythagorean School, where mathematics, science and philosophy were studied. The school developed a brotherhood and performed secret rituals. He and his followers believed that the whole universe was based on numbers.
Pythagoras was murdered when he was 77, and the brotherhood was disbanded.
The square on the hypotenuse in any right-angled triangle is equal to the sum of the squares on the other two sides.
c a b
c a b
That is, or
2 2 2
2 2
= +
= +
ch4.indd 171 7/9/09 12:37:50 AM
Proof
equal corresponding+s
ADC ACB
. x 65
8 06 to 2 decimal places
==
2. Find the exact value of y .
Solution
c a b
y y y y
8 4
64 16
48 48 16 3 4 3
2 2 2
2 2 2
2 2
`
#
= +
= +
= +
=
=
=
=
3. Find the length of the diagonal in a square with sides 6 cm. Answer to 1 decimal place.
Solution
6 cm
6 cm
.
c a b
c
6 6 72
72 8 5
2 2 2
2 2
= +
= +
=
==
So the length of the diagonal is 8.5 cm.
Leave the answer in surd form for the exact answer.
CONTINUED
ch4.indd 173 6/25/09 2:43:08 PM
1. Find the value of all pronumerals, correct to 1 decimal place.
(a)
(b)
(c)
(d)
2. Find the exact value of all pronumerals.
(a)
(b)
(c)
(d)
4.6 Exercises
4. A triangle has sides 5.1 cm, 6.8 cm and 8.5 cm. Prove that the triangle is right angled.
Solution
6.8 cm
8.5 cm 5.1 cm
Let c=8 5. (largest side) and a and b the other two smaller sides.
. .
. .
.
a b
c
c a b
5 1 6 8 72 25 8 5 72 25
2 2 2 2
2 2
2 2 2
`
+ = +
=
=
=
= +
So the triangle is right angled
.
ch4.indd 174 6/25/09 2:43:10 PM
3. Find the slant height s of a cone with diameter 6.8 m and perpendicular height 5.2 m, to 1 decimal place.
4. Find the length of CE , correct to 1 decimal place, in this
rectangular pyramid. AB=8.6 cm and CF=15.9 .cm
5. Prove that DABC is a right-angled triangle.
6. Show that DXYZ is a right-angled isosceles triangle.
X
Y 1 Z
1 2
7. Show that AC=2BC.
8. (a) Find the length of diagonal AC in the fi gure.
Hence, or otherwise, prove (b)
that AC is perpendicular to DC .
9. Find the length of side AB in terms of b .
10. Find the exact ratio of YZ XY in terms of x and y in DXYZ.
ch4.indd 175 6/25/09 2:43:12 PM
11. Show that the distance squared between A and B is given by d2=13t2-180t+625.
12. An 850 mm by 1200 mm gate is to have a diagonal timber brace to give it strength. To what length should the timber be cut, to the nearest mm?
13. A rectangular park has a length of 620 m and a width of 287 m. If I walk diagonally across the park, how far do I walk?
14. The triangular garden bed below is to have a border around it.
How many metres of border are needed, to 1 decimal place?
15. What is the longest length of stick that will fi t into the box below, to 1 decimal place?
16. A ramp is 4.5 m long and 1.3 m high. How far along the ground does the ramp go? Answer correct to one decimal place .
4.5 m
1.3 m
17. The diagonal of a television screen is 72 cm. If the screen is 58 cm high, how wide is it?
18. A property has one side 1.3 km and another 1.1 km as shown with a straight road diagonally through the middle of the property. If the road is 1.5 km long, show that the property is not rectangular.
1.3 km
1.1 km 1.5 km
19. Jodie buys a ladder 2 m long and wants to take it home in the boot of her car. If the boot is 1.2 m by 0.7 m, will the ladder fi t?
ch4.indd 176 6/25/09 2:48:53 PM