Math Problem Statement

List the sides of each triangle in order from shortest to longest. In the triangle given, angle G = 28°, angle F = 110°, and angle H is unknown.

Solution

In the triangle given, you are asked to list the sides in order from shortest to longest. To determine this order, we can use the following principle:

In any triangle, the side opposite the smallest angle is the shortest, and the side opposite the largest angle is the longest.

From the triangle:

  1. Angle G=28G = 28^\circ
  2. Angle F=110F = 110^\circ
  3. Since the sum of angles in a triangle is 180180^\circ, angle H=18028110=42H = 180^\circ - 28^\circ - 110^\circ = 42^\circ

Now, in terms of side lengths:

  • The side opposite angle 2828^\circ (smallest angle) is the shortest.
  • The side opposite angle 110110^\circ (largest angle) is the longest.
  • The side opposite angle 4242^\circ is in between.

So, the order of the sides from shortest to longest is:

  1. GHGH (opposite 2828^\circ)
  2. GFGF (opposite 4242^\circ)
  3. HFHF (opposite 110110^\circ)

Therefore, the answer is: GHGH, GFGF, HFHF (from shortest to longest).

Would you like more detailed explanations on how to apply these principles or any additional assistance?

Here are some questions you might explore next:

  1. What if the triangle had a right angle? How would that affect side lengths?
  2. How do you find the length of a side if you know two sides and the angle between them?
  3. Can you use trigonometry (like the sine or cosine rule) to find exact lengths of sides here?
  4. What would change if two angles in a triangle were equal?
  5. How does the Pythagorean theorem relate to this kind of problem?

Tip: Always remember to add up the angles in a triangle to verify their correctness before deciding side lengths.

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Math Problem Analysis

Mathematical Concepts

Triangle Properties
Angle-Side Relationship

Formulas

Sum of angles in a triangle (180°)
Ordering sides by opposite angles

Theorems

Triangle Inequality Theorem

Suitable Grade Level

Grades 8-10