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? pls help

Answer 1

Given:-

• Both triangles have the same base lengths.

• One triangle has a height that is three times the height of the other triangle.

To Find:-

• Ratio of heights and the areas of the triangles are equal/proportional.

Solution:-

Here, Let the base of both triangles be b.

Height of one triangle be h.

then, height of another triangle will be 3h.

Since, Area of T₁ = $\dfrac{1}{2}\times b \times h$    ------------ (i)

And, Area of T₂ = $\dfrac{1}{2}\times b \times 3h$     --------------- (ii)

Now, From Equation (i) and (ii) we get,

⇒  $\dfrac{T_1}{T_2} = \dfrac{\dfrac{1}{2}\times b \times h}{\dfrac{1}{2}\times b \times 3h}$             [ Cancelling $\dfrac{1}{2}\times b$ from numerator and denominator both ]

⇒  $\dfrac{T_1}{T_2} = \dfrac{h}{3h}$                               [ Cancelling h from numerator and denominator both ]

⇒ $\dfrac{T_1}{T_2} = \dfrac{1}{3}$

Hence, Ratio of height 1 : 3 and Ratio of Areas of triangle 1 :  3 is equal.