Hi Gmat_mission.
First, let's remember that if we have three segments, they can build a triangle if the sum of two of them is greater than the third one.
In our case, we have that the side AB=17. We are not told anything more. Now, let's see the statements.
Statement 1:
(1) The length of side BC is 144.
Now, we know two of the sides of the triangle ABC. According to what we have above, we have that $$CA\le AB+BC\ \ \Rightarrow\ \ CA\le17+144\ \ \Rightarrow\ \ CA\le161$$ $$BC\le AB+CA\ \ \Rightarrow\ \ 144\le17+CA\ \ \Rightarrow\ \ CA\ge127$$ Hence, the length of CA must be a number between 127 and 161.
If AC=145 then we will have that ABC is a right triangle because it satisfies the Pythagoras Theorem: $$AB^2+BC^2=CA^2\ $$ $$17^2+144^2=145^2\ $$ $$289+20736=21025$$ $$21025=21025$$ But if AC=130 for instance, then it won't satisfy the Pythagoras Theorem.
So, this statement is
NOT SUFFICIENT
Statement 2:
(2) The length of side AC is 145.
Now, we know two of the sides of the triangle ABC. According to what we have above, we have that $$BC\le AB+CA\ \ \Rightarrow\ \ BC\le17+145\ \ \Rightarrow\ \ BC\le162$$ $$AC\le AB+BC\ \ \Rightarrow\ \ 145\le17+BC\ \ \Rightarrow\ \ BC\ge128$$ Hence, the length of BC must be a number between 128 and 162.
If AC=144 then we will have that ABC is a right triangle because it satisfies the Pythagoras Theorem: $$AB^2+BC^2=CA^2\ $$ $$17^2+144^2=145^2\ $$ $$289+20736=21025$$ $$21025=21025$$ But if BC=150 for instance, then it won't satisfy the Pythagoras Theorem.
So, this statement is
NOT SUFFICIENT
Statement 1 + Statement 2:
(1) The length of side BC is 144.
(2) The length of side AC is 145.
Using both statements we can conclude that the triangle is a right triangle because it satisfies the Pythagoras Theorem, as shown above.
There using both statements is
SUFFICIENT
Hence, the correct answer is the option
_C_.
I hope it helps you. <i class="em em-sunglasses"></i>
