Transcript Converse of Pythagoras

Converse of
Pythagoras
Geometry
Converse of Pythagoras
In a triangle with sides a, b and c where
c is the largest side, if a2+b2=c2,
then the triangle is right angled at C.
c
a
b
Use the above result to check each
triangle below to see if it is right angled.
6cm
13cm
10cm
5cm
12cm
8cm
Use the result to check each triangle
below to see if it is right angled.
5.3cm
3.7cm
8.6cm
4.2cm
3.5cm
8.2cm
5.1cm
8cm
7.8cm
9.3cm
9.8cm
5.7cm
A rectangular picture frame is to be made.
It is 32cm high and 24.5cm wide as shown in the diagram.
To check that the frame is rectangular,
the diagonal, d, is measured.
It is found to be 39.6cm.
Is the frame rectangular?
d
24.5cm
32cm



7m
24m

25m
Long ago farmers made right angles using a
long rope with knots in
it as shown in the diagram.
Explain how a farmer might check
that a field is rectangular using
the piece of rope shown.
The diagram shows a 32 inch television screen.
The size of a television screen is given as the
length of the diagonal. Is the screen rectangular?
television
screen
32′′
24′′
21.2′′
In the diagram PS=4cm, SR=9cm and SQ = 6cm.
Use Pythagoras to calculate
the lengths of PQ and QR.
Hence prove that triangle PQR is right angles at Q.
Q
6cm
P
4cm
S
9cm
R
A flagpole is 9m long.
A rope holds the
flagpole in position.
The rope is 16m long and is
fixed to the ground 13m from
the foot of the pole.
Is the flagpole vertical?
16m
flag
rope
ground
13m
9m
A pupil has 7 sticks of different lengths as shown.
If he picks three of them and
forms a triangle,
how many different triangles can he form?
How many of these triangles will be right angled?
5cm
8cm
9cm
12cm
13cm
15cm
17cm
Coordinates
Prove that the shape formed by joining the 4
points A(-2,0), B(0,4), C(8,0) and D(6,-4)
is a rectangle.
On square paper plot the points A(0,2), B(6,4)
and C(8,-2). Use Pythagoras to find the length
of the sides of the triangle ABC. Use the
converse of Pythagoras to show that the
triangle ABC is right angled at B.
The diagram shows part of the
structure for supporting a bridge.
It is essential that angle ABC
is right angled if the bridge
is to be supported properly.
Is this the case with
the support shown?
B
55m
A
C
56m
54m
Show that for triangle
PQR to be right angled at Q,
we require c=√ ab.
Q
c
P
a
b
R