- #1

DaniNY

- 4

- 0

A - 17

B - 34

C - 35

D - 70

E - 140

Answer is 35 but how?

Thanks in advance

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In summary: So, the maximum area of a triangle with those dimensions is 70. Thanks for the tip!In summary, the greatest possible area of a triangle with one side of length 7 and the another side of length 10 is 35.

- #1

DaniNY

- 4

- 0

A - 17

B - 34

C - 35

D - 70

E - 140

Answer is 35 but how?

Thanks in advance

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- #2

karush

Gold Member

MHB

- 3,269

- 5

consider it to be a right triangle

- #3

DaniNY

- 4

- 0

but how to know that is the greatest area possible?

- #4

Jameson

Gold Member

MHB

- 4,541

- 13

DaniNY said: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?

- #5

karush

Gold Member

MHB

- 3,269

- 5

- #6

DaniNY

- 4

- 0

- #7

Jameson

Gold Member

MHB

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DaniNY said:

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?

- #8

DaniNY

- 4

- 0

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.Jameson said: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?

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

Thanks so much for your help!

- #9

Opalg

Gold Member

MHB

- 2,778

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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 ... .DaniNY said: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?

To find the greatest possible area of a triangle with given coordinates, you can use the formula A = 1/2 * base * height. In this case, the base and height will be the side lengths of the triangle. You can use the distance formula to calculate the side lengths, and then plug them into the area formula to find the maximum area.

Yes, the formula for finding the area of a triangle is A = 1/2 * base * height. To find the greatest possible area, you will need to calculate the side lengths using the distance formula and then plug them into the area formula.

There are multiple methods and formulas for finding the area of a triangle, such as Heron's formula or using trigonometric functions. However, for a given set of coordinates, the formula A = 1/2 * base * height will give you the maximum area.

The placement of the coordinates can affect the greatest possible area of a triangle as it determines the lengths of the sides of the triangle. For example, if the coordinates are closer together, the sides may be shorter and therefore the maximum area will be smaller compared to coordinates that are further apart.

Yes, there are limitations to finding the greatest possible area of a triangle with given coordinates. The coordinates must form a triangle, meaning that the three points cannot be collinear. Additionally, the coordinates must also be unique and cannot form a degenerate triangle with zero area.

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