MHB SAT Math: Greatest Possible Area of Triangle (7, 10)

[DaniNY]

What is the greatest possible area of a triangle with one side of length 7 and the another side of length 10?Choices:

A - 17
B - 34
C - 35
D - 70
E - 140

Answer is 35 but how?

Thanks in advance

 

[karush]

consider it to be a right triangle

 

[DaniNY]

but how to know that is the greatest area possible?

 

[Jameson]

but how to know that is the greatest area possible?

Hi DaniNY, (Wave)

Welcome to MHB! :)

You can prove this fact using trigonometry and/or calculus but for the SAT I recommend just memorizing this. Some things are worth deriving and some thing are worth memorizing. This one is the latter case in my opinion.

Anyway, once you know that it must be a right triangle how can you find the answer?

 

[karush]

if you rotate the side that is the altitude from 90d then the area can only become smaller because the attitude is smaller

 

[DaniNY]

Thanks for you help! I am actually teaching an SAT course and I just want to make sure I can explain this as clearly as possible

 

[Jameson]

Thanks for you help! I am actually teaching an SAT course and I just want to make sure I can explain this as clearly as possible

Ah, ok. :)

Here is an argument for why this is. The general formula for the area of a triangle can be expressed as: $A = ab \sin(C)$, where $a,b$ are two legs of the triangle and $C$ is the angle between them. Let's also say that $a,b$ represent the two shortest legs.

If you examine some possibilities of the $\sin(C)$ term, you'll see that this value is the biggest for $\sin(90 ^{\circ})$. Try some other values of this and you'll get something less than 1. Does that make sense?

 

[DaniNY]

Ah, ok. :)

Here is an argument for why this is. The general formula for the area of a triangle can be expressed as: $A = ab \sin(C)$, where $a,b$ are two legs of the triangle and $C$ is the angle between them. Let's also say that $a,b$ represent the two shortest legs.

If you examine some possibilities of the $\sin(C)$ term, you'll see that this value is the biggest for $\sin(90 ^{\circ})$. Try some other values of this and you'll get something less than 1. Does that make sense?

Ah yes, that does make more sense...I know trig is not required for the SAT and I think many of my students will be confused. I am going to tell them to remember that maximum area occurs in right triangles.

Any ideas on how I can visually show them this concept?

Thanks so much for your help!

 

[Opalg]

I know trig is not required for the SAT and I think many of my students will be confused. I am going to tell them to remember that maximum area occurs in right triangles.

Any ideas on how I can visually show them this concept?

Being an ignorant foreigner, I know nothing about SAT requirements. But I imagine that students are meant to know that the area of a triangle is half the base times the vertical height. If you are told that the base is 10 and the sloping height is 7, then the way to maximise the area is to make the vertical height as big as possible, namely equal to the sloping height ... .