Problem Solving

Triangle T is in the (x,y) coordinate plane. What is the area, in square coordinate units, of triangle T?

(1) Triangle T is an isosceles right triangle.
(2) Triangle T has 1 side that is longer than each of the other 2 sides, and the midpoints of the other 2 sides are (5,6) and (5,3).

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

ritumaheshwari02 wrote:Triangle T is in the (x,y) coordinate plane. What is the area, in square coordinate units, of triangle T?

(1) Triangle T is an isosceles right triangle.
(2) Triangle T has 1 side that is longer than each of the other 2 sides, and the midpoints of the other 2 sides are (5,6) and (5,3).

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Statement1: Gives no info about the side length. Insufficient
Statement2: Just knowing mid points of two sides will not help. Insufficient

Together: Triangle is an Isosceles right triangle and hyp is longer than either side. Join the mid points of the two sides. The line segment joining mid points of these two sides will be parallel to the third side (In this case Hypotenuse) and half its length.
We can easily get the length of the hyp. and then each of the sides.
Since the sides are perpendicular, Area will be 1/2*S^2 (S being the equal sides of right angled isosceles triangle)

I know the explanation is not very easy and simple to understand.....
Will come back and upload a diagram for better understanding soon...

IMOAnswer is C

nisagl750 wrote: Will come back and upload a diagram for better understanding soon...

According to the diagram....
dist between points b & c = sqrt{ (5-5)^2 + (6-3)^2 }
= sqrt(9)
= 3units

Hence length of the Hyp = 6 units.

Applying Pyth theorem. 2a^2 = 6^2
a = 3 root 2

Area = 1/2 a^2 (Since both sides are equal)
= 1/2 * 18
= 9 square units

Hence Both Statements together
are sufficient. Hence C