Problem 1 · Split side lengths around their midpoint · 3 points
Made by AI and not reviewed by a real person.
A rectangle has a perimeter of 56 cm. Its area increases by 20 cm² if its longer side is shortened by 2 cm and its shorter side is lengthened by 2 cm. What was its original area?
- 196
- 180
- 160
- 176
- 192
Show a hint
- The side lengths keep the same sum. Think about how far each one is from their shared midpoint before and after the change.
- If the original sides are 14 + d and 14 − d, the changed sides are 14 + (d − 2) and 14 − (d − 2). Compare their areas.
Show the answer and solution
Answer: C — 160
The side lengths sum to 28 cm, so their midpoint is 14 cm. Write the original sides as 14 + d and 14 − d. Their area is 196 − d²; after the change it is 196 − (d − 2)². Thus d² − (d − 2)² = 20, giving 4d − 4 = 20 and d = 6. The original sides were 20 cm and 8 cm, so the area was 160 cm². The tempting 180 cm² is the area *after* the change.