If the length of side AB is 17, is triangle ABC a right

[Gmat_mission]

If the length of side AB is 17, is triangle ABC a right triangle?

(1) The length of side BC is 144.
(2) The length of side AC is 145.

[spoiler]OA=C[/spoiler]

Source: Manhattan GMAT

[Gmat_mission]

[spoiler]OA=C[/spoiler]

[Vincen]

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>