Question

the perimeter of this isosceles trapezoid is less than 28.

image of isosceles trapezoid with top base 6, bottom base 10, and legs labeled m

using set - builder notation, write the possible values for the length of the sides labeled m.

Explanation:

Step1: Recall Perimeter Formula

The perimeter \( P \) of a trapezoid is the sum of all its sides. For an isosceles trapezoid with bases \( 6 \) and \( 10 \), and legs \( m \) (two legs, so \( 2m \)), the perimeter is \( P = 6 + 10 + 2m \).

Step2: Set Up Inequality

Given the perimeter is less than \( 28 \), so \( 6 + 10 + 2m < 28 \). Simplify the left side: \( 16 + 2m < 28 \).

Step3: Solve for \( m \)

Subtract \( 16 \) from both sides: \( 2m < 28 - 16 \) → \( 2m < 12 \). Divide by \( 2 \): \( m < 6 \). Also, since \( m \) is a side length, \( m > 0 \) (and by triangle inequality, the sum of the two legs must be greater than the difference of the bases, but here the difference of the bases is \( 10 - 6 = 4 \), so \( 2m > 4 \) → \( m > 2 \)? Wait, no—wait, in a trapezoid, the legs must be long enough to connect the bases. The horizontal difference between the bases is \( (10 - 6)/2 = 2 \), so by Pythagoras, the leg \( m \) must satisfy \( m > 2 \) (since the height is positive, so \( m > 2 \)). Wait, but the problem says "length of the sides labeled \( m \)"—maybe the problem assumes \( m > 0 \), but actually, for a trapezoid, the legs must be longer than the height, but if we just consider the perimeter inequality first:

Wait, original perimeter inequality: \( 6 + 10 + 2m < 28 \) → \( 2m < 12 \) → \( m < 6 \). But also, in a trapezoid, the sum of the legs must be greater than the difference of the bases? Wait, no, the two legs and the difference of the bases form a triangle? Wait, if you drop perpendiculars from the ends of the top base to the bottom base, you get two right triangles with base \( (10 - 6)/2 = 2 \), height \( h \), and leg \( m \). So by Pythagoras, \( m^2 = h^2 + 2^2 \), so \( m > 2 \) (since \( h > 0 \)). So combining, \( 2 < m < 6 \)? Wait, but maybe the problem just considers \( m > 0 \) for simplicity? Wait, no—let's check the problem again. The problem says "length of the sides labeled \( m \)"—maybe the problem expects \( m > 0 \) and \( m < 6 \), but actually, the correct lower bound is \( m > 2 \) (from the right triangle with base \( 2 \)). Wait, but let's re-express:

Wait, the perimeter is \( 6 + 10 + 2m < 28 \) → \( 2m < 12 \) → \( m < 6 \). Also, since the trapezoid must exist, the legs must be longer than the horizontal segment (which is \( 2 \)), so \( m > 2 \). So the correct inequality is \( 2 < m < 6 \). Wait, but maybe the problem ignores the triangle inequality and just uses the perimeter? Wait, the problem says "possible values for the length of the sides labeled \( m \)"—maybe the problem assumes \( m > 0 \), but let's check the initial perimeter:

Wait, if \( m = 0 \), the perimeter is \( 16 \), which is less than \( 28 \), but \( m = 0 \) is not a valid side. If \( m = 2 \), the legs would be equal to the horizontal segment, making the trapezoid a rectangle? No, a rectangle has legs equal to height, but here the horizontal segment is \( 2 \), so if \( m = 2 \), the height would be \( 0 \), which is not a trapezoid. So \( m > 2 \) and \( m < 6 \).

But let's go back to the problem: it says "using set-builder notation". So first, solve the perimeter inequality: \( 6 + 10 + 2m < 28 \) → \( 2m < 12 \) → \( m < 6 \). Also, since \( m \) is a side length, \( m > 0 \), but actually, for the trapezoid to be non-degenerate, \( m > 2 \) (because the difference in the bases is \( 4 \), split into two \( 2 \) segments, so the leg must be longer than \( 2 \) to have a positive height). Wait, but maybe the problem doesn't consider that and just uses \( m > 0 \). Wait, let's check the problem statement again: it says "length of the sides labeled \( m \)"—maybe the problem expects \( m > 0 \) and \( m < 6 \), but actually, the correct lower bound is \( m > 2 \). Wait, no—if \( m = 3 \), which is between \( 2 \) and \( 6 \), that works. If \( m = 1 \), the leg would be shorter than the horizontal segment, making the trapezoid impossible (the height would be imaginary). So the correct inequality is \( 2 < m < 6 \). But let's see the initial perimeter inequality: \( 6 + 10 + 2m < 28 \) → \( m < 6 \). And from the trapezoid's leg length, \( m > 2 \). So combining, \( m \) must satisfy \( 2 < m < 6 \). Wait, but maybe the problem assumes \( m > 0 \), but that's incorrect. Wait, maybe the problem is simpler: just the perimeter inequality, ignoring the trapezoid's geometric constraints (maybe it's a "trapezoid" in name, but just a quadrilateral with two sides \( m \)). So if we take \( m > 0 \) (since length can't be zero or negative), then \( 0 < m < 6 \). But that's a bit ambiguous. Wait, let's re-express:

Perimeter: \( 6 + 10 + 2m < 28 \) → \( 2m < 12 \) → \( m < 6 \). Also, since \( m \) is a side length, \( m > 0 \). So the set-builder notation is \( \{ m \mid 0 < m < 6, m \in \mathbb{R} \} \), but actually, since it's a length, \( m > 2 \) (from the right triangle with base \( 2 \)). Wait, maybe the problem doesn't consider that and just uses \( m > 0 \). Let's proceed with the perimeter inequality first.

Step4: Write Set-Builder Notation

The set of all real numbers \( m \) such that \( m > 0 \) (or \( m > 2 \)) and \( m < 6 \). But since the problem is likely expecting the perimeter inequality solution, we'll use \( \{ m \mid 2 < m < 6 \} \) (correct) or \( \{ m \mid 0 < m < 6 \} \) (if ignoring the geometric constraint). Wait, let's check the calculation again:

Perimeter: \( 6 + 10 + 2m < 28 \) → \( 16 + 2m < 28 \) → \( 2m < 12 \) → \( m < 6 \).

Also, for the trapezoid to have positive height, the leg \( m \) must satisfy \( m > \frac{10 - 6}{2} = 2 \) (since the horizontal projection of each leg is \( 2 \), so by Pythagoras, \( m^2 = h^2 + 2^2 \), so \( m > 2 \) when \( h > 0 \)).

Thus, the possible values of \( m \) are \( 2 < m < 6 \).

Answer:

\(\{ m \mid 2 < m < 6, m \in \mathbb{R} \}\) (or if ignoring the geometric constraint, \(\{ m \mid 0 < m < 6, m \in \mathbb{R} \}\), but the correct one with trapezoid constraints is \(2 < m < 6\)).