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
If the length of side AB is 17, is triangle ABC a right
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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:
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:
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:
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>
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:
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.(1) The length of side BC is 144.
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:
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.(2) The length of side AC is 145.
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:
Using both statements we can conclude that the triangle is a right triangle because it satisfies the Pythagoras Theorem, as shown above.(1) The length of side BC is 144.
(2) The length of side AC is 145.
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>