Working out sides of a triangle given area, and the lenght of the base?
If the area of a triangle is 48, the base length is 16cm, what are lengths of 2 other sides knowing that they’re both the same?
if the other 2 sides are the same, then it is an isosceles triangle.
the height of the triangle is equal to the length of the perpendicular bisector of the base.
A = 0.5*B*H
H = 2A/B
H = 2*48/16
H = 6
now, the perpendicular bisector splits the isosceles triangle into 2 congruent right triangles
hence, the length of the missing side is the hypothenuse, the length of the height is a leg, and half of the base is the other leg.
so,
right triangle with legs of length 8 and 6
c² = a² + b²
c² = 6² + 8²
c² = 36 + 64
c² = 100
c = 10
The length of the other 2 sides is 10.
if the other 2 sides are the same, then it is an isosceles triangle.
the height of the triangle is equal to the length of the perpendicular bisector of the base.
A = 0.5*B*H
H = 2A/B
H = 2*48/16
H = 6
now, the perpendicular bisector splits the isosceles triangle into 2 congruent right triangles
hence, the length of the missing side is the hypothenuse, the length of the height is a leg, and half of the base is the other leg.
so,
right triangle with legs of length 8 and 6
c² = a² + b²
c² = 6² + 8²
c² = 36 + 64
c² = 100
c = 10
The length of the other 2 sides is 10.
References :
area of the triangle is 48.
one base length is 16 cm.
other two sides have same length = x.
————————
0.5*b*h=48
0.5*16*h=48
8h=48
therefore, height of the triangle is 6 cm.
Now you have the height, and it touches the base at the middle. So you can split the base into left and right, each of 8 cm.
now find x, using Pythagoras’s theorem. (you have the height, and the half of the base)
x^2 = 8^2 + 6^2
x^2 = 100
square root of x will give you:
x=10 cm
References :
2 sides of the triangle are equal.So it is an isoceles triangle.
In an isoceles triangle,the altitude to the base bisects it.
This can be easily proved.Visit http://en.allexperts.com/q/Geometry-2060/prove.htm to see the proof.
The area of the triangle=48
Base=16cm
Altitude=h cm
The area of the triangle =(1/2)*Base*Altitude
48=(1/2)*16*h
h=6
Consider any one of the triangles on the either sides of the altitude.
It is right angled triangle.Hence according to pythagoras theorem,
h^2 + (1/2*Base)^2=(Other side)^2
[As mentioned earlier,Altitude bisects the base]
(6)^2+[(1/2)(16)]^2=(Other side)^2
36+64=(Other side)^2
(Other side)^2=100
Other side=10 cm
Hence both the other sides are equal to 10 cm.
Hope this has helped you.
References :