The lateral edge PA of a triangular pyramid ABCP is perpendicular to the base and PA = 2sqrt5 in. The base edges AB = 10 in, AC = 17 in, and BC = 21in. Find the length of the segment PQ if Q ∈ BC and AQ is the altitude of the base.

PQ = √84 = 2√21 in ≈ 9.165 in

Step-by-step explanation:

The base edges AB = 10 in, AC = 17 in, and BC = 21 in

Q ∈ BC and AQ is the altitude of the base.

Let BQ = x in ⇒ CQ = (21 - x) in

ΔAQB a right triangle at Q

∴ AQ² = AB² - BQ² = 10² - x² ⇒ (1)

ΔAQC a right triangle at Q

∴ AQ² = AC² - CQ² = 17² - (21-x) ² ⇒ (2)

Equating (1) and (2)

∴ 10² - x² = 17² - (21-x) ²

∴ 10² - x² = 17² - (21² - 42x + x²)

∴ 10² - x² = 17² - 21² + 42x - x²

∴ 10² - 17² + 21² = 42x

∴ 42x = 252

∴ x = 252/42 = 6

Substitute at (1)

∴ AQ² = AB² - BQ² = 10² - x² = 100 - 36 = 64 = 8²

∴ AQ = 8 in

∵ The lateral edge PA of a triangular pyramid ABCP is perpendicular to the base

∴ PA⊥AQ

∴ ΔPAQ is a right triangle at A,

PA = 2sqrt5 in and AQ = 8 in

∴ PQ (hypotenuse) = √ (PA² + AQ²)

∴ PQ = √[ (2√5) ² + 8²] = √ (20+64) = √84 = 2√21 in ≈ 9.165 in