Question

Two triangles have the same base lengths. One triangle has a height that is three times the height of the other triangle. Are the heights and the areas of the two triangles proportional?

Answer 1

Topic :

• Mensuration

Given :

• Two triangles have same base lengths.
• One triangle has a height that is three times the height of the other triangle.

To Find :

• Whether the heights and areas of the two triangles are proportional ?

Concept Used :

Area of Triangle = ( Base × Height ) / 2

Solution :

Let two triangles be ∆1 and ∆2.

( Assume ∆1 has more height than ∆2. )

Let height of ∆1 be h.

Let height of ∆2 be h'.

Ratio of heights of the triangles,

It is given that,

h = 3h'

h/h' = 3

Ratio of areas of the triangles,

ar.(∆1)/ar.(∆2) = (1/2×Base×h)/(1/2×Base×h')

Canceling (1/2 × Base) from numerator and denominator,

ar.(∆1)/ar.(∆2) = h/h'

ar.(∆1)/ar.(∆2) = 3

As ratio of heights of the triangles and ratio of areas of triangle are equal, we can say heights and areas of the given two triangles are proportional.

Answer :

Yes, the heights and areas of the given two triangles are proportional.