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The lengths of two sides of a triangle are 6 cm and 8 cm. What is the length of the third side of the triangle?
(1) The triangle is a right angle triangle.
(2) The length of the third side of the triangle is an integer.
(1) The triangle is a right angle triangle.
(2) The length of the third side of the triangle is an integer.
06 Mar 2023, 16:34
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Since the sum of any two sides of a triangle needs to be greater than the third side, we have:
\(c + 6 > 8 \Rightarrow c > 2\)
\(c < 6 + 8 \Rightarrow c < 14\)
So, even before checking the statements, we know \(2 < c < 14\)
(I) The triangle is a right angle triangle.
Since it is a right triangle, we can apply Pythogoras. We have two possible hypothenuses: the unknown side of length \(c\) or the side of length \(8\).
First option: \(6^{2} + 8^{2} = c^{2} \Rightarrow c = 10\)
Second option: \(6^{2} + c^{2} = 8^{2} \Rightarrow c = 2\sqrt{7}\)
I is not enough
(II) The length of the third side of the triangle is an integer.
We know \(2 < c < 14\), so there are multiple options for integers in this interval.
II is not enough
(I + II)
With I, we have \(c = 10\) or \(c = 2\sqrt{7}\). With II, we know the length needs to be an integer, so the only possibility is \(c = 10\).
So the answer is \(C\)
\(c + 6 > 8 \Rightarrow c > 2\)
\(c < 6 + 8 \Rightarrow c < 14\)
So, even before checking the statements, we know \(2 < c < 14\)
(I) The triangle is a right angle triangle.
Since it is a right triangle, we can apply Pythogoras. We have two possible hypothenuses: the unknown side of length \(c\) or the side of length \(8\).
First option: \(6^{2} + 8^{2} = c^{2} \Rightarrow c = 10\)
Second option: \(6^{2} + c^{2} = 8^{2} \Rightarrow c = 2\sqrt{7}\)
I is not enough
(II) The length of the third side of the triangle is an integer.
We know \(2 < c < 14\), so there are multiple options for integers in this interval.
II is not enough
(I + II)
With I, we have \(c = 10\) or \(c = 2\sqrt{7}\). With II, we know the length needs to be an integer, so the only possibility is \(c = 10\).
So the answer is \(C\)
08 Mar 2023, 10:39
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Statement one should be enough. it says that the triangle is a right triangle, with two of its sides 6 and 8. This indicates that the triangle is special right triangle with sides (3:4:5) x 2 = 6:8:10. 10 is the 3rd side.
