The lengths of the three sides of a triangle are given as 3cm, 4cm and 5cm. The ratio of their corresponding altitudes to the opposite vertices will be:
- A3:4:5
- B9:8:7
- C20:15:12
- D5:4:3
Solution & Step-by-step Explanation
Let the sides of the triangle be a=3cm, b=4cm, and c=5cm.
Let the corresponding altitudes to these sides be h
a
, h
b
, and h
c
respectively.
The area (Δ) of a triangle can be expressed as:
Δ=
2
1
×base×altitude
Thus, we have:
Δ=
2
1
×a×h
a
=
2
1
×b×h
b
=
2
1
×c×h
c
This implies:
a×h
a
=b×h
b
=c×h
c
=2Δ
Therefore, the altitudes are inversely proportional to the sides:
h
a
:h
b
:h
c
=
a
1
:
b
1
:
c
1
h
a
:h
b
:h
c
=
3
1
:
4
1
:
5
1
To convert this into integer ratios, multiply each term by the LCM of 3,4, and 5, which is 60:
h
a
:h
b
:h
c
=(
3
1
×60):(
4
1
×60):(
5
1
×60)=20:15:12
Let the corresponding altitudes to these sides be h
a
, h
b
, and h
c
respectively.
The area (Δ) of a triangle can be expressed as:
Δ=
2
1
×base×altitude
Thus, we have:
Δ=
2
1
×a×h
a
=
2
1
×b×h
b
=
2
1
×c×h
c
This implies:
a×h
a
=b×h
b
=c×h
c
=2Δ
Therefore, the altitudes are inversely proportional to the sides:
h
a
:h
b
:h
c
=
a
1
:
b
1
:
c
1
h
a
:h
b
:h
c
=
3
1
:
4
1
:
5
1
To convert this into integer ratios, multiply each term by the LCM of 3,4, and 5, which is 60:
h
a
:h
b
:h
c
=(
3
1
×60):(
4
1
×60):(
5
1
×60)=20:15:12