Please tell me how we will take the area

It happens to be a right-angled triangle so you could do [imath]\tfrac{1}{2}[/imath] the base AC [imath]\times[/imath] the height CB
Both of these lengths could be found using Pythagoras' Theorem if you are familiar with it.
 
You have it on a square grid, where you can tell which grid corners the line segments are intersecting,
so you may use Pick's Theorem.

Area = (1/2)b + i - 1, where b is the number of grid corners on the perimeter of the triangle (in this case),
and i is the number of grid corners inside the shape.

Please show your work here if you decide to use this method.
 
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Alternatively for any [imath]\triangle[/imath] you can start at a vertex (corner) e.g. A and write down the directions to get from A to the other points B and C: (using +/- numbers)
1646946958813.png
Then do [imath]\hspace1ex 4 \times 6 \hspace1ex– \hspace1ex2 \times 2[/imath], and half the answer. The size of this answer will be the area.
 
lex said:
Alternatively for any [imath]\triangle[/imath] you can start at a vertex (corner) e.g. A and write down the directions to get from A to the other points B and C: (using +/- numbers)
View attachment 31612
Then do [imath]\hspace1ex 4 \times 6 \hspace1ex– \hspace1ex2 \times 2[/imath], and half the answer. The size of this answer will be the area.
Click to expand...
That is a neat way - which I did not know. Thanks Lex
 
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lex said:
@Subhotosh Khan
Nice, but I'm afraid it's only [imath]\tfrac{1}{2} |\overrightarrow{AB} \times \overrightarrow{AC}| [/imath]
Click to expand...
I realised that - but it is a devious way to introduce matrices and vector cross-product to middle school-students without boring them to death first.

I learned the cross-product trick when I had to calculate the area of a quadrilateral figure in our FEM class.
 
lex said:
Alternatively for any [imath]\triangle[/imath] you can start at a vertex (corner) e.g. A and write down the directions to get from A to the other points B and C: (using +/- numbers)
View attachment 31612
Then do [imath]\hspace1ex 4 \times 6 \hspace1ex– \hspace1ex2 \times 2[/imath], and half the answer. The size of this answer will be the area.
Click to expand...
You can also use the more general "shoelace formula" for area of a polygon:

Art of Problem Solving

artofproblemsolving.com artofproblemsolving.com
en.wikipedia.org

Shoelace formula - Wikipedia

en.wikipedia.org en.wikipedia.org

Applying that full method to our problem, we arrange the coordinates of the three points,

2 1​
4 7​
6 3​

and then multiply diagonally:

|(2*7 + 4*3 + 1*6) - (1*4 + 7*6 + 3*2)|/2 = |32 - 52|/2 = 20/2 = 10​
 
Yes, it was a bit sneaky!
I wondered might it prompt the OP to question about other polygons, and lead us to the shoelace theorem.
I fear we are alone. Oh well!
 
Subhotosh Khan said:
I realised that - but it is a devious way to introduce matrices and vector cross-product to middle school-students without boring them to death first.

I learned the cross-product trick when I had to calculate the area of a quadrilateral figure in our FEM class.
Click to expand...
Dr.Peterson said:
You can also use the more general "shoelace formula" for area of a polygon:
Click to expand...
lex said:
Yes, it was a bit sneaky!
I wondered might it prompt the OP to question about other polygons, and lead us to the shoelace theorem.
I fear we are alone. Oh well!
Click to expand...
While I agree (wholeheartedly) that lex's method is a very slick :thumbup:8-)way of obtaining the answer, I wonder whether it really suits the needs of the OP? Given the facts that the correct option is already indicated and s/he asked "how" it was reached, it seems to me that s/he is trying to gain some understanding of how to solve the problem?

In my answer I was attempting to show how the subtraction of the areas of the 3 coloured triangles from the surrounding rectangle's area would illustrate a way to arrive at the desired result. Of course, that does assume that the OP knows how to calculate the area of both rectangles and right-angled triangles but, if not, then it seems unlikely s/he would have had the competence to submit the post as shown.

If s/he had any comprehension of vector products then I suspect s/he might not have had to ask the question in the first place. :unsure::D

When offering 'help' in response to any (new) member's post I try to think of the simplest way to explain how to approach their problem (unless it's clear that they posses some advanced Math skills) but this is, clearly, hampered, in many cases, by not knowing what their ability level is! :rolleyes:

Perhaps another paragraph might be added to the READ BEFORE POSTING page (preferably at the start) instructing new members to provide some indication of their Mathematical skills/abilities? Their age (if < 20) or their qualifications (exam passes, further/higher education attainments) or even just a brief description of how much "Maths" they know? :unsure:
(Unfortunately, as far as I can glean from my limited time on the forum, so many of them don't appear to bother to read (or understand?) what's asked of them on that page. :))

No criticism of other contributors is intended here; just my thoughts about how best to approach questions where "insufficient information" is provided by the OP. (And a possible way to remedy that?)

Here's something else in a related area for the mathematically-minded to ponder....

Question: What do you get if you cross an elephant with a banana?
Answer: |elephant| |banana| sin(θ) î
 
The Highlander said:
While I agree (wholeheartedly) that @lex's method is a very slick :thumbup:8-)way of obtaining the answer, I wonder whether it really suits the needs of the OP? Given the facts that the correct option is already indicated and s/he asked "how" it was reached, it seems to me that s/he is trying to gain some understanding of how to solve the problem?
Click to expand...
Your method is absolutely the best way; I think we've been adding alternative methods as a conversation among ourselves, since the OP hasn't even looked since the first answer was given.
The Highlander said:
No criticism of other contributors is intended here; just my thoughts about how best to approach questions where "insufficient information" is provided by the OP. (And a possible way to remedy that?)
Click to expand...
There is no remedy. It just happens, and we live with it.
 
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