Does the right triangle has the greatest area?? Can't the included angle be obtuse which would result in a larger area?

I am so incapable of understanding this question and also the solutions above. It is not necessarily a 45-45-90 triangle so isn't there many possibilities?

Would it be possible to give me a detailed explanation of the solution in the problem? I have this "it cannot be determined" answer in mind but obviously I am wrong.

The height or the base of the triangle can be anything, no? Would have been easier to explain what I am thinking with pictures but I guess they're not allowed.

More simple than this is very difficult to figure it out how to explain

But, on what basis can you regard this triangle as a right triangle, you can surely say it as isosceles, and only after drawing a perpendicular to the base, you can say that this is a right triangle.

From question it is an isosceles triangle.

For area to be highest, it has to be a right angled triangle.

1/2 * base *height

1/2 * 2 * 2 = 2 is highest possible area. Any option less than this can be answer option.

Hence all the 4 option.

Please correct if my answer is wrong.

Basic, the more uniform the shape is the higher the area will be. For a triangle with two equal sides the right triangle with 45: 45: 90 got the highest area.

we know that the third side will be |2-2|<x<|2+2| i.e lie between 0-4. So my doubt arises that the area will be between 0 to 2.As the area is between 0-2 will the option D be the part of the answer ? I feel the areas will only be 0.5 1 and 1.5 the area can be 1.999 too but not 2 is what I feel. Please correct me if I'm wrong I have my gre in 4 days

why we assume it is a right triangle?
The question never mention about the height and if they asking about something else they will put it for you.
yeah it can be an isosceles triangle with different angle, but if that was the case how are we going to calculate the area?

PLEASE IF I'M WRONG CORRECT ME THIS WILL HELP ME A LOT

See my explanation above, please.

Regards

Property of triangle:
Sum of 2 sides must be greater than 3rd side:
side1=2, side2=2
side1+side2 = 4
means third side must be less than 4

Another property of triangle:
Difference of 2 sides must be less than 3rd side
side1=2, side=2
side1-side2 = 2-2 =0
means third side must be greater than zero

The value will lie between 0 - 2.

I have no idea where im going wrong, please correct me.

so I draw a height from A to the base, call it h
call BC=b

we know that 0<b<4

so I took 2 extremes:

1) b=0.1

by Pythagoras, h= sqrt(2^2 - (0.1/2)^2) = 2

so minimum area is 0.09 (approx. equal to 0)

2) b=3.9

h = sqrt (2^2 - (3.9/2)^2) = 0.44

maximum area= 3.9*0.44/2= 0.9

I knowwww that something must be wrong can someone help out pleaseee

There is a property that the right angled triangle has the largest area.
Now, we are given that two sides are equal. If two sides are equal then the angles opposite them will also be equal.
So we get a 45-45-90 right angled triangle.
Now, the base and perpendicular = 2.
So, the area = \(\frac{1}{2} \times 2 \times 2 = 2\)

2 is the largest area we can get.
Now, if the third side = 0, then essentially we get a line of length 2. and the area of triangle becomes 0.

So, the minimum area has to be greater than 0. and maximum area is 2.

The values of area possible lie between 0 and 2.

\(0 < Area < 2\)

OA, A,B,C,D

Hi, I understand everything above until you said that "the value will lie between 0-2". can you explain how you got that

Hi!

The length of the 3rd side should be less than the sum of the other two sides and greater than the difference between the two sides
So, lets say AC is the third side.

2+2=4
2-2=0

Thus, 0<AC<4
Thus AC lies between 0 and 4. It could be any value, not necessarily integer value.

Hope this helps!

Among all, Right Angle triangle has maximum area. Hence area can have maximum of 2*2/2 i.e 2. below this , all values are possible.

The easiest way to understand this question is to draw out (or visualise) a right angle triangle with both sides AB and AC as the legs and the third side BC as the hypotenuse.

The area of the triangle in this scenario will be

(1/2)BH = (1/2)*2*2 = 2

If we imagine AB as the base and AC as the height, we can visualize the AC changing as the angle BAC widens or reduces.

This helps us to retain AB as the base and have a varied height.

In both cases with a widened angle BAC or reduced BAC, the height reduces and can reduce to almost zero as the angle BAC gets to either 0 degrees or 180 degrees.

Since it is a triangle, the area cannot be 0, the area is above 0 and less than or equal to 2.

Hence, 0 > Area >= 2

The answers A to D are in this range.

Why 2 is included in answer if answer lies between 0-2?