Square Composed of Triangles
Algebra
Level
3
A square composed of five triangles is given with unknown side lengths. Four triangle perimeters are given. What is the perimeter of the 5th triangle, P5=? Give your answer to the nearest 1/100th.
The answer is 23.44.
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P1 & P3 are congruent triangles, while P2 is an isosceles right triangle. Therefore, P1's hypothenuse
= P3's hypothenuse
= 26 / (1 + 1 + √2)
= 26 / (2 + √2)
= 7.62
By this, X + Y
= { P1 (or P3), the perimeter } - 7.62
= 10.
P2's hypothenuse
= [ 26 / (1 + 1 + √2) ] × √2
= 26√2 / (2 + √2)
= 26 / (1 + √2)
= 10.77 = √[ (Y + X)² + (Y - X)² ]
(Y - X)² = 10.77² - 10²
= 0.77 × 20.77
= 15.99
≈ 16
Y - X = √15.99 ≈ 4
Y + X = 10
2Y = 14 ==> Y = 7 & X = 3
Answer, P5
= X + (Y + X) + √[ X² + (Y + X)² ]
= 3 + 10 + √[ 3² + (10)² ]
= 13 + √109
= 23.4403065
P/s : Those without a calculator can get √109 by using P4 and P2's hypothenuse, with 28.21 - Y - 10.77 = 28.21 - 7 - 10.77 = 28.21 - 17.77 = 10.44