Pythagoras’ Theorem
A right-angled triangle has legs 6 cm and 8 cm.
Find the hypotenuse.
c2=62+82=36+64=100
c=10c = 10c=10 cm
A triangle has sides 7 cm, 24 cm and 25 cm.
Is it right-angled?
72+242=49+576=625
252=62525^2 = 625252=625
Yes, it is right-angled.
A right-angled triangle has a 30∘ angle and hypotenuse 14 cm.
Find the shorter side.
Shorter side = 12×14=7\frac{1}{2} \times 14 = 721×14=7 cm
A rectangle has length 9 cm and width 12 cm.
Find the diagonal.
d2=92+122=81+144=225
d=15d = 15d=15 cm
Two triangles are similar.
The smaller triangle has sides 5 cm, 6 cm and 7 cm.
The corresponding side to 5 cm in the larger triangle is 10 cm.
Find the other sides.
Scale factor = 2
Sides = 12 cm and 14 cm
The hypotenuse of a right-angled triangle is 13 cm and one side is 5 cm.
Find the other side.
x2=132−52=169−25=144
x=12x = 12x=12 cm
The sides of a triangle are 6 cm, 8 cm and 9 cm.
Is the triangle right-angled?
62+82=36+64=100
92=819^2 = 8192=81
Not right-angled.
A right-angled triangle has sides in the ratio
1:3:21 : \sqrt{3} : 21:3:2.
If the shortest side is 5 cm, find the hypotenuse.
Scale factor = 5
Hypotenuse = 2×5=102 \times 5 = 102×5=10 cm
A square has diagonal 10 cm.
Find the length of one side.
x2=10
x=52x = 5\sqrt{2}x=52 cm
AB/DE=4/7
If AB=8 cm, find DE.
DE=4/7×8=14 cm
A right-angled triangle has sides 9 cm, x cm and 15 cm.
Find x.
x2=152−92=225−81=144
x=12x = 12x=12 cm
A triangle has sides 20\sqrt{20}20, 4 and 6.
Determine whether it is right-angled.
(20)\sqrt2+42=20+16=36
62=366^2 = 3662=36
Yes, it is right-angled.
A right-angled triangle has angles 45∘,45∘,90∘45^\circ, 45^\circ, 90^\circ45∘,45∘,90∘.
One leg is 6 cm.
Find the hypotenuse.
c=6 2\sqrt cm
Two perpendicular paths are 15 m and 20 m long.
Find the shortest distance between their ends.
d2=152+202=225+400=625
d=25d = 25d=25 m
Two triangles have angles
50∘,60∘,70∘ and
60∘,50∘,x
Find x and decide if the triangles are similar.
x=70∘
Yes, triangles are similar.
A ladder reaches a wall at a height of 12 m.
The foot of the ladder is 5 m from the wall.
Find the length of the ladder.
L2=122+52=144+25=169
L=13L = 13L=13 m
A triangle has sides 9 cm, 12 cm and 15 cm.
Show whether it is right-angled.
92+122=81+144=225
152=22515^2 = 225152=225
Right-angled triangle.
The hypotenuse of a 30∘ − 60∘ − 90∘ triangle is 18 cm.
Find the side opposite 60∘60^\circ60∘.
Side = 939\sqrt{3}93 cm
A rectangle has diagonal 26 cm and one side 10 cm.
Find the other side.
x2=262−102=676−100=576
x=24x = 24x=24 cm
One triangle has angles 35∘ and 65∘.
Another triangle has angles 65∘and 80∘.
Are the triangles similar?
First triangle third angle = 80∘
Second triangle third angle = 35∘
Yes, similar.
A right-angled triangle has area 30 cm² and one leg is 5 cm.
Find the hypotenuse.
1/2×5×x=30
x=12
c2=52+122=25+144=169
c=13c = 13c=13 cm
A triangle has sides 10 cm, 10 cm and 14 cm.
Can this triangle be right-angled?
102+102=200
142=19614^2 = 196142=196
Not right-angled.
A right-angled triangle has equal legs and area 32 cm².
Find the length of each leg.
1\2 x x2/sqrt=32
x2/sqrt=64
x=8 cm
Two right-angled triangles are similar.
One has legs 8 cm and 15 cm.
The corresponding leg in the second triangle is 16 cm.
Find the other leg.
Scale factor = 2
Other leg = 15×2=3015 \times 2 = 3015×2=30 cm
Two similar triangles have perimeters 24 cm and 36 cm.
A side in the smaller triangle is 6 cm.
Find the corresponding side in the larger triangle.
Scale factor = 36/24=3/2
Side = 6×3/2=9 cm