Asked by Elijah
Can the sides of a triangle have lengths 8, 18, and 20
All Answers 1
Answered by
GPT-5 mini
AI
Yes. Check the triangle inequalities:
- 8 + 18 = 26 > 20
- 8 + 20 = 28 > 18
- 18 + 20 = 38 > 8
So a nondegenerate triangle with sides 8, 18, 20 exists.
Extras:
- It's obtuse, since 8^2 + 18^2 = 388 < 20^2 = 400 (angle opposite side 20 is > 90°). In fact cos(angle opposite 20) = (8^2+18^2-20^2)/(2·8·18) = −1/24.
- Area (Heron): s = (8+18+20)/2 = 23, area = sqrt(23·15·5·3) = 15·sqrt(23) ≈ 71.93.
- 8 + 18 = 26 > 20
- 8 + 20 = 28 > 18
- 18 + 20 = 38 > 8
So a nondegenerate triangle with sides 8, 18, 20 exists.
Extras:
- It's obtuse, since 8^2 + 18^2 = 388 < 20^2 = 400 (angle opposite side 20 is > 90°). In fact cos(angle opposite 20) = (8^2+18^2-20^2)/(2·8·18) = −1/24.
- Area (Heron): s = (8+18+20)/2 = 23, area = sqrt(23·15·5·3) = 15·sqrt(23) ≈ 71.93.
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