the question that this triangle is right-angled tells us that it will be easier to simply find the two-sides adjacent to the right-angle and then use the triangle area formula
A
=
1
2
b
h
on them - area = half base x height.
Find side lengths: show right-angled by Pythagoras' Theorem
Pythagoras' Theorem tells us that in a right-angled triangle (and only in right-angled triangles), the three side lengths
a
b
c
relate with
a
2
+
b
2
=
c
2
. So showing that the lengths of the sides here fit this formula will tell us that the triangle is right-angled. Use the distance formula between two points in 3D (which is itself a simple application of Pythagoras' Theorem) to work the side lengths out:
d
=
√
(
x
1
−
x
2
)
2
+
(
y
1
−
y
2
)
2
+
(
z
1
−
z
2
)
2
Length of side
A
B
−−− :
A
B
−−−
=
√
(
4
−
2
)
2
+
(
7
−
1
)
2
+
(
9
−
6
)
2
=
√
4
+
36
+
9
=
√
49
=
7
Length of side
A
C
−−− :
A
C
−−−
=
√
(
8
−
2
)
2
+
(
5
−
1
)
2
+
(
−
6
−
6
)
2
=
√
36
+
16
+
144
=
√
196
=
14
Length of side
B
C
−−− :
B
C
−−−
=
√
(
8
−
4
)
2
+
(
5
−
7
)
2
+
(
−
6
−
9
)
2
=
√
16
+
4
+
225
=
√
245
=
7
√
5
Looking at the squares of the side lengths, we see that indeed
A
B
−−−
2
+
A
C
−−−
2
=
B
C
−−−
2
. The long side is
B
C
−−− , so this is the hypotenuse, and the right-angle is in the corner of the triangle at point
A
.
