a square ana an equilateral triangle have equal perimeters . if the diagonal of the square is 12 underroot 2 cm , then area of the triangle is : 64 underroot 3 cm square , 24 underroot 2 cm square , 48 underroot 3 cm square , 24 underroot 3 cm square
Answers
Answer:
Step-by-step explanation:Let
A
s
be the area of the square,
P
s
be the perimeter of the square and
a
s
be the length of a side of the square. (All sides have equal lengths.)
Let
At be the area of the triangle, Pt
be the perimeter of the triangle and
at be the length of a side of the triangle. (All sides have equal lengths.)
============================================
1) As we know the length of the diagonal of the square, we can compute the length of a side of the square using the Pythagoras formula:
a2s+a2s=d2⇒2a2s=(12√2)
2⇒a2s=122
⇒
a
s
=
12
cm
2) Knowing the length of one side of the square (and thus knowing all lengths of a square), we can easily compute the square's perimeter:
P
s
=
12
⋅
4
=
48
cm
3) We know that the square and the equilateral triangle have the same perimeter, thus
P
t
=
48
cm
4) As all sides have the same length in an equilateral triangle, the length of one side is
a
t
=
P
t
3
=
16 cm
5) Now, to compute the area of the equilateral triangle, we need the height
h
which can be computed with the Pythagoras formula again:
h
2
+
(
a
t
2
)
2
=
a
2
t
⇒
h
2
+
8
2
=
16
2
⇒
h
2
=
192
=
64
⋅
3
⇒
h
=
8
√
3
cm
6) At last, we can compute the area of the triangle:
A
t
=
1
2
⋅
h
⋅
a
t
=
1
2
⋅
8
√
3
⋅
16
=
64
√
3
cm
2
