Ratio of the surfaces of two bases of a frustum equals ratio of one of their appropriate sides squared ?

[Ognjen]

I attached the photo of the frustum given in the problem.

Since the bases are similar triangles, and the problem specifies that the ratio of appropriate sides of the bases is 1/2, solution says the following equation holds:

[math]B/B_1 = AB^2/A_1B_1^2 => B/B_1 = 2^2 / 1^2[/math]
I don't understand how this can be if area of a triangle can only be expressed ( in general case ) with one side and its adequate altitude ( unless we can incorporate a larger number of variables such as with using Hero's formula, or if we have radii of the inscribed or circumscribed circles ). Thus, I see no justification for the exponentiation of the sides AB and A_1B_1.

To my mind, the only mathematically justified way to write the equation using ratios would be:

[math]B/B_1 = AB*h_A / A_1B_1*h_a[/math]
But this is useless, since we have no ratio between appropriate heights of any two sides given in the problem.

I didn't give the whole formulation of the problem because this fraction was the only one I had any trouble with.

I would immensely appreciate any help.

 

[JeffM]

I attached the photo of the frustum given in the problem.

Since the bases are similar triangles, and the problem specifies that the ratio of appropriate sides of the bases is 1/2, solution says the following equation holds:

[math]B/B_1 = AB^2/A_1B_1^2 => B/B_1 = 2^2 / 1^2[/math]
I don't understand how this can be if area of a triangle can only be expressed ( in general case ) with one side and its adequate altitude ( unless we can incorporate a larger number of variables such as with using Hero's formula, or if we have radii of the inscribed or circumscribed circles ). Thus, I see no justification for the exponentiation of the sides AB and A_1B_1.

To my mind, the only mathematically justified way to write the equation using ratios would be:

[math]B/B_1 = AB*h_A / A_1B_1*h_a[/math]
But this is useless, since we have no ratio between appropriate heights of any two sides given in the problem.

I didn't give the whole formulation of the problem because this fraction was the only one I had any trouble with.

I would immensely appreciate any help.

The notation is awful. A, B, etc. are labels of points rather than numbers. You can‘t square them or divide them.

You have explicitly said that you have not given us all the information provided in the problem. Therefore, there may be aspects of the problem that you have erroneously deemed irrelevant. If so, we cannot possibly help you.

However, I suspect that this is as simple as this.

Given that the two triangles are similar, the length of the heights, [imath]h_1[/imath] and [imath]h_2[/imath] are in the same proportion as the lengths of the sides, [imath]s_1[/imath]and [imath]s_2[/imath]. The question is what is the proportion between the areas of the two triangle, [imath]a_1[/imath] and [imath]a_2[/imath]?



[math]\dfrac{s_1}{s_2} = p = \dfrac{h_1}{h_2}.\\ a_1 = \dfrac{s_1h_1}{2}.\\ a_2 = \dfrac{s_2h_2}{2}.\\ \therefore \ \dfrac{a_1}{a_2} = \dfrac{\dfrac{s_1h_1}{2}}{\dfrac{s_2h_2}{2}}= \dfrac{s_1}{s_2} * \dfrac{h_1}{h_2} = p * p = p^2.[/math]

 

[Ognjen]

The notation is awful. A, B, etc. are labels of points rather than numbers. You can‘t square them or divide them.

You have explicitly said that you have not given us all the information provided in the problem. Therefore, there may be aspects of the problem that you have erroneously deemed irrelevant. If so, we cannot possibly help you.

However, I suspect that this is as simple as this.

Given that the two triangles are similar, the length of the heights, [imath]h_1[/imath] and [imath]h_2[/imath] are in the same proportion as the lengths of the sides, [math]s_1[/imath]and [imath]s_2[/imath]. The question is what is the proportion between the areas of the two triangle, [imath]a_1[/imath] and [imath]a_2[/imath]? [math]\dfrac{s_1}{s_2} = p = \dfrac{h_1}{h_2}.\\ a_1 = \dfrac{s_1h_1}{2}.\\ a_2 = \dfrac{s_2h_2}{2}.\\ \therefore \ \dfrac{a_1}{a_2} = \dfrac{\dfrac{s_1h_1}{2}}{\dfrac{s_2h_2}{2}}= \dfrac{s_1}{s_2} * \dfrac{h_1}{h_2} = p * p = p^2.[/math]

Thank you so much, I get it now

The notation difficulties resulted from the official notation of the workbook I found the problem in. I didn't want to change anything for the sake of consistency and for fear of modifying something crucial whose relevance I wasn't aware of.