A Pythagorean triad is a set of three positive integers, a, b and c, for which c2= a2+ b2. That is, they are integral dimensions of a right-angled triangle. All multiples of Pythagorean triads are also triads. For example, we know that {3, 4, 5} is a Pythagorean triad. Hence, {6, 8, 10}, {9, 12, 15} and {12, 16, 20} are also Pythagorean triads.
■ The converse of Pythagoras’ theorem
The converse of Pythagoras’ theorem can be used to show that a triangle is right angled.
Example 1
State Pythagoras’ theorem for this triangle using:
a side notation b angle notation Solutions
a p2= q2+ r2 b QR2= PQ2+ PR2
5.3 Pythagoras’ theorem
In any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
c2= a2+ b2 a c
b
A1
A3 5 4
3 A2
A1 A2 A3
A1 A2 A3
If the square on one side of a triangle is equal to the sum of the squares on the other two sides, then the angle between the two shorter sides is a right angle.
EG+S p
R P
Q r
q
Example 2
Determine whether the following are Pythagorean triads.
a 5, 12, 13 b 7, 8, 9
Solutions
a 132= 52+ 122 b 92= 72+ 82
169 = 25 + 144 81 = 49 + 64
169 = 169 81 = 113
This is a true statement, This is not a true statement,
∴ 5, 12, 13 is a Pythagorean triad. ∴ 7, 8, 9 is not a Pythagorean triad.
Example 3
Find the value of the pronumeral in each diagram. Give your answer correct to 1 decimal place where necessary.
a b
Solutions
a a2= 122+ 352 b w2+ 112= 192
= 144 + 1225 w2+ 121 = 361
= 1369 −121 −121
∴ a = w2= 240
= 37 ∴w =
= 15.5 (1 decimal place)
1 Write down Pythagoras’ theorem for each triangle using:
i side notation ii angle notation
a b
EG+S
EG+S
35 cm
12 cm
a cm
19 cm w cm 11 cm
1369
240
Exercise 5.3
q
R Q
P
r
p
F g E
G
e f
2 Write true (T) or false (F) for each of these.
a XY2= XW2+ WY2 b XW2= WZ2+ XZ2 c YZ2= ZW2+ WY2 d WY2= XW2+ XY2 e YZ2= WY2+ XZ2 f XZ2= XY2+ YZ2
3 Use Pythagoras’ theorem to determine which of the following triangles, not drawn to scale, are right-angled. If the triangle is right-angled, name the hypotenuse.
a b
4 Which of the following are Pythagorean triads?
a 6, 8, 10 b 3, 5, 7 c 12, 35, 37 d 10, 15, 20
5 Explain, without calculation, why 2, 3, 6 could not be a Pythagorean triad.
6 Find the value of the pronumeral in each of the following. Give your answers correct to 1 decimal place.
a b
c d
■ Consolidation
7 Find the value of x in each of these, correct to 1 decimal place.
a b
Y
Z
X W
X Z
Y
6 7
5 R T
S 17
8 15
5 mm 11 mm
x mm
p mm
12 mm
7 mm
23 mm 28 mm
z mm
a mm 36 mm
69 mm
x cm
22 cm
10 mm x mm
8 a In ∆QRS, ∠R = 90°, QR = 25 km and RS = 38 km. Find the length of QS, correct to 4 significant figures.
b In ∆ABC, ∠A = 90°, AC = 41 cm and BC = 75 cm. Find the length of AB, correct to the nearest centimetre.
9 Find, correct to 1 decimal place, the possible lengths of the third side of a right-angled triangle in which two of the sides measure 6 m and 14.4 m.
10 a A ladder reaches 7.5 m up a wall and the foot of the ladder is 2.4 m away from the base of the wall. Find the length of the ladder correct to the nearest centimetre.
b A ladder of length 4.5 m leans against a wall. The foot of the ladder is 1.2 m away from the base of the wall. How far up the wall does the ladder reach, correct to 1 decimal place?
11 a A ship sailed 6 km due north, then changed course and sailed 14 km due east. How far is the ship from its starting point?
b Emilia drove due east from J to K, then turned and drove 15 km due south to L. If L is 48 km from J, find how far east Emilia drove.
12 While out orienteering, a group of students walked 350 m due west, 290 m due north then 560 m due east. How far, to the nearest metre, are the students from their starting point?
13 a A non-right-angled isosceles triangle has a base length of 66 cm and a height of 56 cm.
How long are the equal sides?
b A non-right-angled isosceles triangle has equal sides of length 75 mm and a height of 21 mm. How long is the base?
14 Find the value of d in each figure.
15 Find the length of the chord UV in this circle.
a 5.8 km b
8 km
12.2 km d km
7.7 km
10.1 km
8.5 km d km
7.2 cm 3.9 cm V
U O W
16 Find the values of the pronumerals in each of these. Answer correct to 1 decimal place where necessary.
17 In ∆PQR, S is a point on PR such that QS is
perpendicular to PR.
If QS= 12 cm, PQ = 15 cm and QR = 20 cm,
20 Find the length of the longest rod that will fit
completely inside this rectangular prism.
■ Further applications
21 In the diagram, AB= 35 mm,
DE= 15 mm and AE = 130 mm.
a Find the length of BD.
b Find the length of CD if
22 If a, b, c are the sides of a right-angled triangle and c is the hypotenuse, prove that any multiple of a, b, c will also be the sides of a right-angled triangle.
23 The expressions p2− q2, 2pq, p2+ q2, where p and q are positive integers and p> q can be used to generate Pythagorean triads. By substituting values for p and q, find at least 5 Pythagorean triads.
24 Two air force jets took off from the same airport at 3 pm. One jet flew due south at 320 km/h while the other flew due west at 370 km/h. How far apart are the jets at 5:30 pm if each maintains the same course and speed? (Answer correct to the nearest whole kilometre.) 25 The Mountain Top Ski Resort is situated on top of a 3.6 km high mountain. A cable car
from the resort travels along the cable at 5 m/s and takes 13 min to reach the ground station.
a How long is the cable?
b How far is the ground station from the foot of the mountain?
The perimeter of a two-dimensional figure is the total distance around its boundary.
The formulae below can also be used to find the perimeter of some common figures.