In the rectangular coordinate system above, the area of triangular

So area of Trapezium = 1/2 * sum of parallel sides * height
= 1/2 * (OQ + O'R) * 7
= 49/2
=24.5

Now from this we need to subtract the two tingles to give area of triangle QPR
= 24.5 - ( 6*2)
= 12.5

Answer A.

Multiple methods exist to solving this problem.

The easiest, at least for me, was pythag triples. See my diagram

We can actually deduce the distance of each point using the x-y axis, giving us 5,5 (legs of the triangle).

How do we know that PGR is the right triangle? It is also can be isosceles?

I FOUND THE EASIEST AND QUICKEST METHOD EVER .It took me 1 minute to solve .

step 1 write down all the coordinates in vertical format inside MODE (because area has to be positive always ) and multiply it with 1/2

1/2 * MODE/ X1 - X2 X1-X3 /MODE here 1/2* MODE / 0 - 7 0-4 / MODE
................/ Y1 - Y2 Y1-Y3 /....................................../ 3 - 4 3-0 /

Step2 (upon solving you get ) 1/2* / A B / SO cross multiply you get 1/2 * /AD - BC / . Here, 1/2 * MODE / -7 -4/
.................................................../ C D /........................................................................................... /-1 -3/

Your answer is 25/2 = 12.5 . This method is quickest , just be careful with signs .

TO UNDERSTAND IT IN DETAILS WATCH IT ON BELOW LINK
https://www.youtube.com/watch?v=It9Vd3UFYVg

No need to calculate anything here. The area of the largest triangle inscribed in a rectangle is half the area of the rectangle. Here we have a rectangle of area 7*4 = 28. The largest inscribed triangle happens to be if Q = (0.4) ,i.e base of triangle = length of rectangle. Since the area of the largest triangle possible is 14, the answer has to be less than 14. I.e A.

How do we know that the triangle is right? Could it not have been an isosceles triangle?

I believe it's answered on the previous pages. Please re-read the discussion. Hope it helps.

How can we state that the triangle in this problem is a right angle triangle?

use the determinant method 1/2{(x1-x2)(y2-y3)-(x2-x3)(y1-y2)}.this is the easiest method and the modular sign has to be used becs area cant be negative

I have found that these questions are easier to solve using the Discriminant property.

Set all X's and Y's as separate columns in the matrix, with a column of 1s, solve for the 3x3 discriminant and appropriately multiply by +/- 1/2.

This is the easiest way to do this problem... do not look any further Team.

You need to memorize your pythagorean triples (3,4,5) (5,12,13) and if you don't have these memorized to so now or else you will be in very deep trouble. Memorize (3,4,5) & (5,12,13)

It is very easy to determine the triangle centerQP which will give a hypotenuse of 5 (height of Point Q is (0,3), length of Point P is (0,4), your hypotenuse is easily 5.
This same concept goes line QR being they hypotenuse of 5 for another triangle that you can visualize.

Now, this is the part that messed me up initially-- do not try to find the hypotenuse of triangle QPR... its asking for the Area. finding the hyp will result in 5squareroot2 which is a similar answer but A=1/2bh and is sufficient to find with the 1/2 5*5

Hope this helps.

Great explanation JeffTargetTestPrep. In calculation triangle area, how to decide using which value to minus which value between P (4,0) , R(7,4) for e.g? Thanks

Can we find out the coordinates of the point on the line QR where a Perpendicular from P, if drawn would meet? Then that would be our Height for the triangle, no matter if it's right-angled or not!!

Hi BrentGMATPrepNow, to clarify how do we work out base * height values from the given cocordinate points and do we ignore minus sign if getting a negative value ? Thanks Brent

If you get a negative value, then turn it into a positive value

Get it thanks BrentGMATPrepNow
Also how do we work out base * height values from the given cocordinate points?
Lastly for rectangular area of l * W, is it just using R (7,4) here? Thanks Brent

For this question you have to figure out the base and the height of each triangle.
The top left triangle has a horizontal base with length 7 and a height of 1, which means the area is 3.5
The bottom left triangle has a horizontal base with length 4 and a height of 3, which means the area is 6
The bottom right triangle has a horizontal base with length 4 and a height of 4, which means the area is 6