## lunes, 23 de enero de 2017

### 1128. Alpha. Beta. Gamma. Delta.

The four internal angles of a quadrilateral, measured in degrees, are natural numbers and are in geometric progression: alpha, beta, gamma and delta. Calculate alpha, beta and delta when gamma is...

a) ... 81°
b) ... 90°
c) ... 96°

Pepe Chapuzas solved this problem (these three problems). He found the natural solutions and the other solutions...

Dear Teacher:
The sum of the four internal angles of a quadrilateral equals 360°

alpha + beta + gamma + delta = 360°

If  R  is the common ratio of the geometric progression...

gamma/R2 + gamma/R + gamma + gamma·R = 360
gamma + gamma·R + (gamma360)·R2 + gamma·R3 = 0
Then...

a)
81R3  279R2 + 81R + 81 = 0
R = 3
alpha = 81°/9 = 9°
beta = 81°/3 = 27°
delta = 3 · 81° = 243°
b)
90R3  270R2 + 90R + 90 = 0
R = 1
alpha = beta = gamma = delta = 90°

The progression is constant and the quadrilateral is a square.

c)
96R3  264R2 + 96R + 96 = 0
R = 2
alpha = 96°/4 = 24°
beta = 96°/2 = 48°
delta = 2 · 96° = 192°

The other solutions are irrational (even negative).

a)

b)

c)

#### 1 comentario:

1. Good ole Pepe - he usually gets there 😏