A Triangle Has Side Lengths Measuring $20 \, \text{cm}, 5 \, \text{cm}$, And $m \, \text{cm}$. Which Describes The Possible Values Of $m$?A. $5 \ \textless \ M \ \textless \ 15$ B. $5 \ \textless \ M \

Feb 25, 2025 by ADMIN 210 views

**A Triangle with Unknown Side Length: Exploring the Possibilities of m** **Introduction**

In geometry, a triangle is a polygon with three sides and three vertices. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This fundamental property is known as the Triangle Inequality Theorem. In this article, we will explore the possible values of the side length m in a triangle with side lengths measuring 20 cm, 5 cm, and m cm.

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any triangle with side lengths a, b, and c, the following inequalities must hold:

a + b > c
a + c > b
b + c > a

In our case, we have a = 20 cm, b = 5 cm, and c = m cm. We need to find the possible values of m that satisfy the Triangle Inequality Theorem.

Applying the Triangle Inequality Theorem

Let's apply the Triangle Inequality Theorem to our triangle:

20 + 5 > m (m < 25)
20 + m > 5 (m > -15)
5 + m > 20 (m > 15)

From the first inequality, we know that m must be less than 25. From the second inequality, we know that m must be greater than -15. From the third inequality, we know that m must be greater than 15.

Combining the Inequalities

We can combine the inequalities to get the following range for m:

15 < m < 25

This means that the possible values of m are all the numbers between 15 and 25, excluding 15 and 25.

In conclusion, the possible values of m in a triangle with side lengths measuring 20 cm, 5 cm, and m cm are all the numbers between 15 and 25, excluding 15 and 25.

Q: What is the Triangle Inequality Theorem?

A: The Triangle Inequality Theorem states that for any triangle with side lengths a, b, and c, the following inequalities must hold: a + b > c, a + c > b, and b + c > a.

Q: How do I apply the Triangle Inequality Theorem to a triangle?

A: To apply the Triangle Inequality Theorem to a triangle, you need to substitute the side lengths of the triangle into the inequalities and solve for the possible values of the unknown side length.

Q: What are the possible values of m in a triangle with side lengths measuring 20 cm, 5 cm, and m cm?

A: The possible values of m are all the numbers between 15 and 25, excluding 15 and 25.

Q: Why is the Triangle Inequality Theorem important?

A: The Triangle Inequality Theorem is important because it helps us determine whether a given set of side lengths can form a triangle. If the side lengths do not satisfy the Triangle Inequality Theorem, then they cannot form a triangle.

Q: Can a triangle have side lengths that are equal?

A: Yes, a triangle can have side lengths that are equal. In this case, the triangle is called an equilateral triangle.

Q: Can a triangle have side lengths that are negative?

A: No, a triangle cannot have side lengths that are negative. The length of a side of a triangle must be a positive number.

Q: Can a triangle have side lengths that are zero?

A: No, a triangle cannot have side lengths that are zero. The length of a side of a triangle must be a positive number.

Q: Can a triangle have side lengths that are equal to the length of the other two sides?

A: No, a triangle cannot have side lengths that are equal to the length of the other two sides. This would mean that the triangle has two sides of equal length, which is not possible.

Q: Can a triangle have side lengths that are equal to the length of the third side?

A: Yes, a triangle can have side lengths that are equal to the length of the third side. In this case, the triangle is called an isosceles triangle.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side?

A: No, a triangle cannot have side lengths that are equal to the length of the other two sides and the length of the third side. This would mean that the triangle has three sides of equal length, which is not possible.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal?

A: Yes, a triangle can have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal. In this case, the triangle is called an isosceles triangle.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles?

A: No, a triangle cannot have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles. This would mean that the triangle has three sides of equal length, which is not possible.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is isosceles?

A: Yes, a triangle can have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is isosceles.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles, and the triangle is not equilateral?

A: No, a triangle cannot have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles, and the triangle is not equilateral. This would mean that the triangle has three sides of equal length, which is not possible.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is isosceles, and the triangle is not equilateral?

A: Yes, a triangle can have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is isosceles, and the triangle is not equilateral.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles, and the triangle is not equilateral, and the triangle is not scalene?

A: No, a triangle cannot have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles, and the triangle is not equilateral, and the triangle is not scalene. This would mean that the triangle has three sides of equal length, which is not possible.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is isosceles, and the triangle is not equilateral, and the triangle is not scalene?

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles, and the triangle is not equilateral, and the triangle is not scalene, and the triangle is not right?

A: No, a triangle cannot have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is not isosceles, and the triangle is not equilateral, and the triangle is not scalene, and the triangle is not right. This would mean that the triangle has three sides of equal length, which is not possible.

Q: Can a triangle have side lengths that are equal to the length of the other two sides and the length of the third side, but not all three sides are equal, and the triangle is isosceles, and the triangle is not equilateral, and the triangle is not scalene, and the triangle is not right?