A triangle has side lengths of 4, 5, and 6. Which of the following correctly classifies this triangle.
Acute, Right, Obtuse, Isosceles.
To classify a triangle, we can use the triangle inequality theorem and the Pythagorean theorem.
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's check if the given triangle satisfies this condition:
- The sum of the lengths of the sides 4 and 5 is 4 + 5 = 9, which is greater than the length of the third side 6.
- The sum of the lengths of the sides 4 and 6 is 4 + 6 = 10, which is greater than the length of the third side 5.
- The sum of the lengths of the sides 5 and 6 is 5 + 6 = 11, which is greater than the length of the third side 4.
Since all three conditions are satisfied, we can confirm that the given triangle is valid.
Now, let's use the Pythagorean theorem to determine the type of triangle:
- If a^2 + b^2 = c^2, where a and b are the lengths of the two shorter sides, and c is the length of the longest side, then the triangle is a right triangle.
- If a^2 + b^2 < c^2, then the triangle is an obtuse triangle.
- If a^2 + b^2 > c^2, then the triangle is an acute triangle.
For this triangle, the side lengths are 4, 5, and 6. Assuming 6 is the longest side, we have:
4^2 + 5^2 = 16 + 25 = 41
6^2 = 36
Since 41 is greater than 36, we can conclude that the triangle is an acute triangle.
Therefore, the correct classification for this triangle is Acute.
To classify a triangle based on its side lengths, we first need to determine if it is an isosceles triangle. An isosceles triangle is a triangle that has two sides of equal length.
Given the side lengths of the triangle are 4, 5, and 6, none of the sides are equal, so the triangle cannot be isosceles.
Next, let's determine if the triangle is a right triangle. A right triangle is a triangle that has one angle equal to 90 degrees. We can use the Pythagorean theorem to check if the triangle satisfies this condition.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the side lengths are 4, 5, and 6. Since 4^2 + 5^2 = 16 + 25 = 41 and 6^2 = 36, the triangle does not satisfy the Pythagorean theorem. Therefore, it is not a right triangle.
Finally, let's classify the triangle based on its angles. For a triangle with side lengths of 4, 5, and 6, we can use the Law of Cosines to determine the angles of the triangle.
Using the Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C), where c is the side opposite angle C.
For this triangle, c = 6, a = 4, and b = 5. Plugging these values into the equation and solving for cos(C), we get:
36 = 16 + 25 - 40 * cos(C)
16 = 40 * cos(C)
cos(C) = 16/40
cos(C) = 0.4
Using the inverse cosine function, cos^(-1), we can find the angle C:
C = cos^(-1)(0.4)
C = 66.42 degrees
Since we know one angle is greater than 90 degrees (66.42 degrees), the triangle is classified as an obtuse triangle.
Therefore, the correct classification for this triangle is "Obtuse."
https://www.google.com/search?source=hp&q=triangle+4+5+6&oq=triangle+4%2C+5.+&gs_l=psy-ab.1.0.0i22i30k1l4.2007.30153.0.32712.25.20.5.0.0.0.120.1737.18j2.20.0....0...1.1.64.psy-ab..0.25.1772...0j46j0i131k1j0i46k1j0i10k1.k4KuVLV74rY
the longest side is opposite the largest angle
since 4^2 + 5^2 > 6^2
... the largest angle is < 90º
... because 6 is to short to be a hypotenuse
so ALL the angles are < 90º ... acute