529/HK16. problem / Math - Solved Math Problems

MR-529 / 14. problem

Length of the sides of cyclic trapezoid are: AB = 5 cm, BC = 2,5 cm, CD = 2 cm and DA = 2,5 cm.

a) Calculate the angles of the trapezoid!
b) Determine the ratio of ABC and ACD triangles!
c) The inner angles of the trapezoid are marked with an arc of 5 mm radius. Calculate the sum of the four arc lengths!

Source: Magyarország: Unified final exam for high schools - Mathematics - intermediate level - 03.05.2016. - 14..)

Algebra → Exponential equations and inequalities → Exponential equations

a) B T = \frac{5 - 2}{2} = 1, 5

∆ B C T \to \cos β = \frac{B T}{B C} = \frac{1, 5}{2, 5} = 0, 6

β = a r c \cos 0, 6

β = 53, 13 °

α = β = 53, 13 °

α + γ = 180 °

γ = 180 ° - α = 180 ° - 53, 13 °

γ = 126, 87 °

δ = γ = 126, 87 °

b) ∆ B C T \to 2, 5^{2} = m^{2} + 1, 5^{2}

m^{2} = 2, 5^{2} - 1, 5^{2}

m^{2} = {(\frac{5}{2})}^{2} - {(\frac{3}{2})}^{2} = \frac{25}{4} - \frac{9}{4} = \frac{16}{4} = 4

m = 2

T_{A B C} = \frac{A B \cdot m}{2} = \frac{5 \cdot 2}{2} = 5 c m^{2}

T_{A C D} = \frac{C D \cdot m}{2} = \frac{2 \cdot 2}{2} = 2 c m^{2}

\frac{T_{A B C}}{T_{A C D}} = \frac{5}{2}

c) α + β + γ + δ = 360 ° \to ○

l = 2 r π = 2 \cdot 0, 5 \cdot π

l = π c m

Ez tudnod kell igen

a) α = β = 53, 13 °, δ = γ = 126, 87 ° b) \frac{T_{A B C}}{T_{A C D}} = \frac{5}{2} c) l = π c m