Asked by Alexis
How would you decompose this trapezoid to find its area? Find the area of the trapezoid.
Convince Me!
How is finding the area of the trapezoid in Example 1 different from finding the area of the trapezoid in the Try It!
All Answers 1
Answered by
GPT-5 mini
AI
I don't see the picture or numbers you refer to, so I’ll give a general decomposition method (and a short derivation of the trapezoid area formula) and explain how two typical trapezoid problems can differ. If you post the specific trapezoid from Example 1 / the Try It!, I’ll compute its area exactly.
General decomposition (and derivation)
- Let the two parallel sides (bases) have lengths b1 (top) and b2 (bottom), and let the perpendicular distance between them (the height) be h.
- Drop two perpendiculars from the endpoints of the top base down to the bottom base. That splits the trapezoid into:
- a central rectangle of width b1 and height h, and
- two right triangles (one at each side) whose combined base length is b2 − b1 and whose heights are both h.
- Area = area(rectangle) + area(two triangles)
= b1·h + 1/2·(b2 − b1)·h
= [b1 + (b2 − b1)/2]·h
= (b1 + b2)/2 · h.
So the area of any trapezoid = (average of the bases) × height = (b1 + b2)/2 · h.
Alternative viewpoint (rearrangement)
- Cut off one of the end triangles and slide it to the other side; the trapezoid becomes a parallelogram whose base equals (b1 + b2)/2 and whose height is still h. Area = base · height = (b1 + b2)/2 · h.
Short numeric example
- If b1 = 8, b2 = 14, and h = 5, area = (8 + 14)/2 · 5 = 11 · 5 = 55 square units.
How Example 1 might differ from the Try It!
- Often a textbook “Example 1” shows a right trapezoid (one or both nonparallel sides perpendicular to the bases). In that case you can see the height directly on the figure and decompose into a rectangle and two right triangles with easily known leg lengths.
- A “Try It!” might show an oblique trapezoid (non-right legs). The decomposition idea is the same, but you may first need to:
- compute the height (drop perpendiculars and use similar triangles or Pythagorean theorem), or
- compute the areas of non-right triangles, or
- cut and rearrange a triangle to form a parallelogram.
- In short: the underlying area formula and decomposition idea are the same; the difference is whether you can read the height and triangle bases directly (easy) or must compute them from other given lengths (requires extra steps).
If you paste the two trapezoid figures (or give b1, b2, and h), I’ll show the exact decomposition and compute the area for each.
General decomposition (and derivation)
- Let the two parallel sides (bases) have lengths b1 (top) and b2 (bottom), and let the perpendicular distance between them (the height) be h.
- Drop two perpendiculars from the endpoints of the top base down to the bottom base. That splits the trapezoid into:
- a central rectangle of width b1 and height h, and
- two right triangles (one at each side) whose combined base length is b2 − b1 and whose heights are both h.
- Area = area(rectangle) + area(two triangles)
= b1·h + 1/2·(b2 − b1)·h
= [b1 + (b2 − b1)/2]·h
= (b1 + b2)/2 · h.
So the area of any trapezoid = (average of the bases) × height = (b1 + b2)/2 · h.
Alternative viewpoint (rearrangement)
- Cut off one of the end triangles and slide it to the other side; the trapezoid becomes a parallelogram whose base equals (b1 + b2)/2 and whose height is still h. Area = base · height = (b1 + b2)/2 · h.
Short numeric example
- If b1 = 8, b2 = 14, and h = 5, area = (8 + 14)/2 · 5 = 11 · 5 = 55 square units.
How Example 1 might differ from the Try It!
- Often a textbook “Example 1” shows a right trapezoid (one or both nonparallel sides perpendicular to the bases). In that case you can see the height directly on the figure and decompose into a rectangle and two right triangles with easily known leg lengths.
- A “Try It!” might show an oblique trapezoid (non-right legs). The decomposition idea is the same, but you may first need to:
- compute the height (drop perpendiculars and use similar triangles or Pythagorean theorem), or
- compute the areas of non-right triangles, or
- cut and rearrange a triangle to form a parallelogram.
- In short: the underlying area formula and decomposition idea are the same; the difference is whether you can read the height and triangle bases directly (easy) or must compute them from other given lengths (requires extra steps).
If you paste the two trapezoid figures (or give b1, b2, and h), I’ll show the exact decomposition and compute the area for each.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.