The Coordinates Of The Vertices Of $\triangle R S T$ Are $R(-3,1), S(-1,4$\], And $T(3,1$\].Which Statement Correctly Describes Whether $\triangle R S T$ Is A Right Triangle?A. $\triangle R S T$ Is A Right

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Introduction

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In geometry, a right triangle is a triangle in which one of the angles is a right angle, which is equal to 90 degrees. The coordinates of the vertices of a triangle can be used to determine whether it is a right triangle or not. In this article, we will discuss how to determine whether a triangle with given coordinates is a right triangle.

The Formula for the Distance Between Two Points

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To determine whether a triangle is a right triangle, we need to calculate the lengths of its sides. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be calculated using the formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Calculating the Lengths of the Sides of $\triangle R S T$

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The coordinates of the vertices of $\triangle R S T$ are $R(-3,1), S(-1,4)$, and $T(3,1)$. We can use the formula above to calculate the lengths of the sides of the triangle.

The Length of Side $R S$

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The length of side $R S$ can be calculated as follows:

$$d_{RS} = \sqrt{(-1 - (-3))^2 + (4 - 1)^2}$$

$$d_{RS} = \sqrt{(2)^2 + (3)^2}$$

$$d_{RS} = \sqrt{4 + 9}$$

$$d_{RS} = \sqrt{13}$$

The Length of Side $S T$

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The length of side $S T$ can be calculated as follows:

$$d_{ST} = \sqrt{(3 - (-1))^2 + (1 - 4)^2}$$

$$d_{ST} = \sqrt{(4)^2 + (-3)^2}$$

$$d_{ST} = \sqrt{16 + 9}$$

$$d_{ST} = \sqrt{25}$$

$$d_{ST} = 5$$

The Length of Side $T R$

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The length of side $T R$ can be calculated as follows:

$$d_{TR} = \sqrt{(-3 - 3)^2 + (1 - 1)^2}$$

$$d_{TR} = \sqrt{(-6)^2 + 0^2}$$

$$d_{TR} = \sqrt{36}$$

$$d_{TR} = 6$$

Determining Whether $\triangle R S T$ is a Right Triangle

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To determine whether $\triangle R S T$ is a right triangle, we need to check if the square of the length of one of the sides is equal to the sum of the squares of the lengths of the other two sides. This is known as the Pythagorean theorem.

Checking the Pythagorean Theorem for $\triangle R S T$

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We can check the Pythagorean theorem for $\triangle R S T$ by calculating the square of the length of each side and checking if the sum of the squares of the lengths of the other two sides is equal to the square of the length of the third side.

Checking the Pythagorean Theorem for Side $R S$

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The square of the length of side $R S$ is:

$$d_{RS}^2 = (\sqrt{13})^2$$

$$d_{RS}^2 = 13$$

The sum of the squares of the lengths of the other two sides is:

$$d_{ST}^2 + d_{TR}^2 = 5^2 + 6^2$$

$$d_{ST}^2 + d_{TR}^2 = 25 + 36$$

$$d_{ST}^2 + d_{TR}^2 = 61$$

Since $d_{RS}^2 \neq d_{ST}^2 + d_{TR}^2$, we can conclude that $\triangle R S T$ is not a right triangle.

Checking the Pythagorean Theorem for Side $S T$

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The square of the length of side $S T$ is:

$$d_{ST}^2 = 5^2$$

$$d_{ST}^2 = 25$$

The sum of the squares of the lengths of the other two sides is:

$$d_{RS}^2 + d_{TR}^2 = (\sqrt{13})^2 + 6^2$$

$$d_{RS}^2 + d_{TR}^2 = 13 + 36$$

$$d_{RS}^2 + d_{TR}^2 = 49$$

Since $d_{ST}^2 \neq d_{RS}^2 + d_{TR}^2$, we can conclude that $\triangle R S T$ is not a right triangle.

Checking the Pythagorean Theorem for Side $T R$

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The square of the length of side $T R$ is:

$$d_{TR}^2 = 6^2$$

$$d_{TR}^2 = 36$$

The sum of the squares of the lengths of the other two sides is:

$$d_{RS}^2 + d_{ST}^2 = (\sqrt{13})^2 + 5^2$$

$$d_{RS}^2 + d_{ST}^2 = 13 + 25$$

$$d_{RS}^2 + d_{ST}^2 = 38$$

Since $d_{TR}^2 \neq d_{RS}^2 + d_{ST}^2$, we can conclude that $\triangle R S T$ is not a right triangle.

Conclusion

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In conclusion, we have determined that $\triangle R S T$ is not a right triangle. The square of the length of each side is not equal to the sum of the squares of the lengths of the other two sides, which means that the Pythagorean theorem does not hold for this triangle.

Final Answer

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The final answer is: $\boxed{0}$

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Q: What is a right triangle?

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A: A right triangle is a triangle in which one of the angles is a right angle, which is equal to 90 degrees.

Q: How do you determine whether a triangle is a right triangle?

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A: To determine whether a triangle is a right triangle, you can use the Pythagorean theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: What is the Pythagorean theorem?

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A: The Pythagorean theorem is a mathematical formula that states:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the lengths of the two sides that form the right angle, and $c$ is the length of the hypotenuse.

Q: How do you use the Pythagorean theorem to determine whether a triangle is a right triangle?

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A: To use the Pythagorean theorem, you need to calculate the square of the length of each side and check if the sum of the squares of the lengths of the other two sides is equal to the square of the length of the third side.

Q: What are the coordinates of the vertices of a right triangle?

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A: The coordinates of the vertices of a right triangle can be used to determine whether it is a right triangle or not. The coordinates of the vertices of a right triangle are usually given as $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$.

Q: How do you calculate the length of a side of a right triangle?

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A: To calculate the length of a side of a right triangle, you can use the distance formula, which is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Q: What is the difference between a right triangle and an oblique triangle?

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A: A right triangle is a triangle in which one of the angles is a right angle, while an oblique triangle is a triangle in which none of the angles are right angles.

Q: Can a triangle have more than one right angle?

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A: No, a triangle cannot have more than one right angle. If a triangle has more than one right angle, it is not a triangle.

Q: Can a right triangle have all sides of equal length?

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A: No, a right triangle cannot have all sides of equal length. If a triangle has all sides of equal length, it is an equilateral triangle, not a right triangle.

Q: Can a right triangle have all angles of equal measure?

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A: No, a right triangle cannot have all angles of equal measure. If a triangle has all angles of equal measure, it is an equiangular triangle, not a right triangle.

Q: Can a right triangle be isosceles?

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A: Yes, a right triangle can be isosceles. An isosceles right triangle is a right triangle in which two of the sides are of equal length.

Q: Can a right triangle be equilateral?

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A: No, a right triangle cannot be equilateral. An equilateral triangle is a triangle in which all three sides are of equal length, and a right triangle cannot have all sides of equal length.

Q: Can a right triangle be obtuse?

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A: No, a right triangle cannot be obtuse. An obtuse triangle is a triangle in which one of the angles is greater than 90 degrees, and a right triangle has one angle that is exactly 90 degrees.

Q: Can a right triangle be acute?

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A: Yes, a right triangle can be acute. An acute triangle is a triangle in which all three angles are less than 90 degrees, and a right triangle has one angle that is exactly 90 degrees.

Q: Can a right triangle be scalene?

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A: Yes, a right triangle can be scalene. A scalene triangle is a triangle in which all three sides are of different lengths, and a right triangle can have all three sides of different lengths.

Q: Can a right triangle be a degenerate triangle?

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A: No, a right triangle cannot be a degenerate triangle. A degenerate triangle is a triangle in which all three vertices lie on the same line, and a right triangle has three vertices that do not lie on the same line.

Q: Can a right triangle be a non-degenerate triangle?

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A: Yes, a right triangle can be a non-degenerate triangle. A non-degenerate triangle is a triangle in which the three vertices do not lie on the same line, and a right triangle has three vertices that do not lie on the same line.

Q: Can a right triangle be a triangle with a negative area?

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A: No, a right triangle cannot be a triangle with a negative area. The area of a triangle is always positive, and a right triangle has a positive area.

Q: Can a right triangle be a triangle with a zero area?

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A: No, a right triangle cannot be a triangle with a zero area. The area of a triangle is always positive, and a right triangle has a positive area.

Q: Can a right triangle be a triangle with an infinite area?

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A: No, a right triangle cannot be a triangle with an infinite area. The area of a triangle is always finite, and a right triangle has a finite area.

Q: Can a right triangle be a triangle with a negative perimeter?

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A: No, a right triangle cannot be a triangle with a negative perimeter. The perimeter of a triangle is always positive, and a right triangle has a positive perimeter.

Q: Can a right triangle be a triangle with a zero perimeter?

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A: No, a right triangle cannot be a triangle with a zero perimeter. The perimeter of a triangle is always positive, and a right triangle has a positive perimeter.

Q: Can a right triangle be a triangle with an infinite perimeter?

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A: No, a right triangle cannot be a triangle with an infinite perimeter. The perimeter of a triangle is always finite, and a right triangle has a finite perimeter.

Q: Can a right triangle be a triangle with a negative semiperimeter?

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A: No, a right triangle cannot be a triangle with a negative semiperimeter. The semiperimeter of a triangle is always positive, and a right triangle has a positive semiperimeter.

Q: Can a right triangle be a triangle with a zero semiperimeter?

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A: No, a right triangle cannot be a triangle with a zero semiperimeter. The semiperimeter of a triangle is always positive, and a right triangle has a positive semiperimeter.

Q: Can a right triangle be a triangle with an infinite semiperimeter?

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A: No, a right triangle cannot be a triangle with an infinite semiperimeter. The semiperimeter of a triangle is always finite, and a right triangle has a finite semiperimeter.

Q: Can a right triangle be a triangle with a negative inradius?

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A: No, a right triangle cannot be a triangle with a negative inradius. The inradius of a triangle is always positive, and a right triangle has a positive inradius.

Q: Can a right triangle be a triangle with a zero inradius?

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A: No, a right triangle cannot be a triangle with a zero inradius. The inradius of a triangle is always positive, and a right triangle has a positive inradius.

Q: Can a right triangle be a triangle with an infinite inradius?

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A: No, a right triangle cannot be a triangle with an infinite inradius. The inradius of a triangle is always finite, and a right triangle has a finite inradius.

Q: Can a right triangle be a triangle with a negative circumradius?

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A: No, a right triangle cannot be a triangle with a negative circumradius. The circumradius of a triangle is always positive, and a right triangle has a positive circumradius.

Q: Can a right triangle be a triangle with a zero circumradius?

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A: No, a right triangle cannot be a triangle with a zero circumradius. The circumradius of a triangle is always positive, and a right triangle has a positive circumradius.

Q: Can a right triangle be a triangle with an infinite circumradius?

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A: No, a right triangle cannot be a triangle with an infinite circumradius. The circumradius of a triangle is always finite, and a right triangle has a finite circumradius.

Q: Can a right triangle be a triangle with a negative height?

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A: No, a right triangle cannot be a triangle with a negative height. The height of a triangle is always positive, and a right triangle has a positive height.

Q: Can a right triangle be a triangle with a zero height?

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A: No, a right triangle cannot be a triangle with a zero height. The height of a triangle is always positive, and a right triangle has a positive height.

Q: Can a right triangle be a triangle with an infinite height?