Trojuholníky 5

Trojuholníky ABC a XYZ sú podobné. Zisti chýbajúce dĺžky strán trojuholníkov.

a) a = 5 cm b = 8 cm
x = 7,5 cm z = 9 cm

b) a = 9 cm c = 12 cm
y = 10 cm z = 8 cm

c) b = 4 cm c = 8 cm
x = 4,5 cm z = 6 cm

Správna odpoveď:

c₁ =  6 cm
y₁ =  12 cm
b₂ =  15 cm
x₂ =  6 cm
a₃ =  6 cm
y₃ =  3 cm

Postup správneho riešenia:

$$\begin{gathered} a_1=5\text{cm} \\ {b}_1=8\text{cm} \\ {x}_1=7,5\text{cm} \\ {z}_1=9\text{cm} \\ {k}_1=x_1/a_1=7,5/5=\frac{3}{2}=1,5 \\ {k}_1=z_1/c_1 \\ {c}_1=z_1/k_1=z_1/\frac{3}{2}=9/1,5=6\text{cm} \end{gathered}$$

$$\begin{gathered} k_1=y_1/b_1 \\ {y}_1=k_1\cdot b_1=\frac{3}{2}\cdot b_1=1,5\cdot 8=12\text{cm} \end{gathered}$$

$$\begin{gathered} -------------------- \\ {a}_2=9\text{cm} \\ {c}_2=12\text{cm} \\ {y}_2=10\text{cm} \\ {z}_2=8\text{cm} \\ {k}_2=z_2/c_2=8/12=\frac{2}{3}\doteq 0,6667 \\ {k}_2=y_2/b_2 \\ {b}_2=y_2/k_2=y_2/\frac{2}{3}=10/0,6667=15\text{cm} \end{gathered}$$

$$\begin{gathered} k_2=x_2/a_2 \\ {x}_2=k_2\cdot a_2=\frac{2}{3}\cdot a_2=0,6667\cdot 9=6\text{cm} \end{gathered}$$

$$\begin{gathered} -------------------- \\ {b}_3=4\text{cm} \\ {c}_3=8\text{cm} \\ {x}_3=4,5\text{cm} \\ {z}_3=6\text{cm} \\ {k}_3=z_3/c_3=6/8=\frac{3}{4}=0,75 \\ {k}_3=x_3/a_3 \\ {a}_3=x_3/k_3=x_3/\frac{3}{4}=4,5/0,75=6\text{cm} \end{gathered}$$

$$\begin{gathered} k_3=y_3/b_3 \\ {y}_3=k_3\cdot b_3=\frac{3}{4}\cdot b_3=0,75\cdot 4=3\text{cm} \end{gathered}$$

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