Which of the following statements about triangle PQR shown in the xy-p

Question

https://gmatclub.com/forum/download/file.php?id=51989 Which of the following statements about triangle PQR shown in the xy-plane are true? Indicate all such statements. I. PQR is a right triangle. II. The area of PQR, is \frac{15}{2} . III. PQR is an isosceles triangle. A. I only B. II only C. III only D. I and II only E. I, II, and III 1.jpg

zhanbo · Answer

My Answer is  (D) , i.e. I and II only

PQ^2 = 50
QR^2 = 5
PR^2 = 45

Because 50 = 5 + 45, PQR is a right triangle.
Now, its area is 1/2 * SQRT(5)*SQRT(45) = 15/2 = 7.5

kanishaksharma · Answer

Answer is D,
To check I
you can calculate slope of QR and PR and if their multiplication comes to be -1 then they are perpendicular to each other:
slope of QR= (4-3)/(2-4) = -1/2
slope of PR = (3-(-3))/4-1= 2
slopeQR* slopePR = -1

to find area II calculate lengths of PR and QR using Pythagoras,
PR= root(36+9)=root(45)
QR= root(4+1)=root(5)
area = 1/2 * PR * QR= 15/2

GMATWhizTeam · Answer

Solution:

Distance  QR = \sqrt{(4-2)^2 + (3-4)^2}

= \sqrt{2^2+(-1)^2}

= \sqrt{5}

Similarly, side  PR = \sqrt{(4-1)^2 + (3-(-3)^2}

= \sqrt{3^2+6^2}

= \sqrt{45}

And, side  PQ = \sqrt{(2-1)^2 + (4-(-3)^2}

= \sqrt{1^2+7^2}

= \sqrt{50}

Now that we have all 3 sides, we see that triangle PQR is a right angle triangle because it follows the Pythagoras theorem:  (\sqrt{50})^2=(\sqrt{45})^2+(\sqrt{5})^2 . This means statement 1 must be true. We can eliminate options B and C.

The area of triangle PQR will be  = \frac{1}{2}	imes \sqrt{45} 	imes \sqrt{5} = \frac{15}{2} . This means statement 2 must also be true. We can eliminate option A.

We can clearly see that triangle PQR is not isosceles triangle. This means statement 3 must not be true.

Hence the right answer is    Option D   .

Crytiocanalyst · Answer

the side lengths are respectively : rt5 , rt45 , rt50 therefore right angled

Area = 1/2 * 15 
Not isosceles therefore IMO D

Which of the following statements about triangle PQR shown in the xy-p

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Which of the following statements about triangle PQR shown in the xy-p

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