Geomeetrilisi nuputamisülesandeid

Antud on kolm ruutu. Kui suur on märgitud nurk?

α=?

Suurima ja väikseima poolringi keskpunktid ühtivad. Kui suur on sinise ala pindala?

2 ühikut

Olgu\ väikseima\ poolringi\ raadiuseks\ r\ ⇒\ selle\ poolringi\ pindala\ on\ S_r=\frac{πr²}{2}\
Suurima\ poolringi\ raadiuse\ saame\ arvutada\ Pythagorase\ teoreemiga\ ⇒\ R=\sqrt{4+r²}\
Seega\ suurima\ poolringi\ pindala\ on\ S_R=\frac{π·\left( \sqrt{4+r²}  \right)² }{2}=\frac{π·\left(  4+r² \right) }{2}\
Kahe\ keskmise\ suurusega\ poolringi\ pindala\ on\ sama\ suured,\ sest\ nende\
 diameetrid\ on\ võrdsed.\ Seega\ tähistame\ nende\ pindala\ A.\
Sinise\ ala\ pindala\ on\ S_s=(S_R-A)+(A-S_r)=S_R-S_r\
S_s=\frac{π·\left(  4+r² \right) }{2}-\frac{πr²}{2}=\frac{4π+πr²-πr²}{2}=2π

Lillad ristkülikud on võrdsed. Kui suure osa moodustab roheline ala musta ruudu pindalast?

x

Olgu\ ristküliku\ lühem\ külg\ x.\ Jagades\ musta\ ruudu\ neljaks\ võrdseks\
 täisnurkseks\ kolmnurgaks,\ saame,\ et\ ristküliku\ pikem\ külg\ on\ 2x.\
Seega\ rohelise\ ala\ pindala\ on\ S_k=2·\frac{x·2x}{2}+\frac{x·\frac{x}{2}}{2}=\frac{9x²}{4}\
Musta\ ruudu\ külg\ on\ ristküliku\ diagonaal\ ⇒\ a=\sqrt{x²+4x²}=\sqrt{5x²}\
Musta\ ruudu\ pindala\ on\ S_m=\left( \sqrt{5x²}  \right)²=5x²\
Roheline\ ala\ moodustab\ mustast\ ruudust\ \frac{\frac{9x²}{4}}{5x²}=\frac{9}{20} 

Inspiratsioon:
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https://twitter.com/Cshearer41
https://drive.google.com/file/d/1hVP8tLURVDphmHsphz5BQLVzHCeTts29/view

a = 1 (slider from 1 to 10, step 1)
A = Point(yAxis)
B = (0.02, -5.26)
hulknurk1, f, g, h, i, C, D = Polygon(A, B, 4)
E = Point(i)
j = Line(B, E)
k = OrthogonalLine(D, j)
F = Intersect(j, k)
l = Line(A, F)
hulknurk2, m, n, p, q, G, H = Polygon(D, F, 4)
I = Point(j)
hulknurk3, r_{2}, s, t, b, J, K = Polygon(B, I, 4)
c = Line(J, I)
L = Intersect(l, c)
d = Segment(F, L)
e = Circle(G, F)
f_1 = Line(H, G)
M = Intersect(l, f_1)
β = Angle(I, L, F)
γ = Angle(G, M, F)
δ = Angle(F, G, M)
g_1 = Segment(H, M)
i_1 = Segment(F, B)
N = Point(c)
α = Angle(F, L, N)
h_1 = Segment(F, M)
j_1 = Segment(I, L)
tekst3: "Võtame\ ringi\ raadiuseks\ ruudu\ külje.\ Ühendades\ kahe\ ristuva\ raadiuse\\
 otspunktid,\ näeme,\ et\ tekkinud\ täisnurkse\ võrdhaarse\ kolmnurga\\
 hüpotenuus\ ühtib\ märgitud\ nurga\ haaraga.\ Saame\ sarnased\ kolmnurgad,\\
 mille\ alusnurgad\ on\ 45°.\\
Kõrvunurkade\ reegli\ järgi\ on\ ∠α\ suuruseks\ seega\\
\\
α=180°-45°=" + (LaTeX[α]) + " 
  "
O = (1, -2)
P = (1, -8)
c_1 = Circle(O, P)
k_1 = Line(P, O)
Q = Intersect(c_1, k_1, 2)
R_{1} = Point(k_1)
d_1 = Circle(O, R_{1})
S = Intersect(d_1, k_1, 2)
T = Midpoint(S, P)
U = Midpoint(R_{1}, Q)
e_1 = CircleSector(O, Q, P)
p_1 = CircleSector(T, P, S)
q_1 = CircleSector(U, Q, R_{1})
r_1 = CircleSector(O, R_{1}, S)
l_1 = OrthogonalLine(S, k_1)
V = Intersect(e_1, l_1)
m_1 = Segment(V, S)
n_1 = Segment(Q, P)
R = Segment(O, V)
r = Segment(S, O)
W = (-4, 3)
Z = (-4, -6)
hulknurk4, s_1, t_1, a_1, b_1, A_1, B_1 = Polygon(W, Z, 4)
c_2 = Circle(W, b_1 / 3)
d_2 = Circle(Z, b_1 / 3)
e_2 = Circle(A_1, b_1 / 3)
f_2 = Circle(B_1, b_1 / 3)
C_1 = Intersect(c_2, b_1, 1)
D_1 = Intersect(c_2, s_1, 1)
E_1 = Intersect(d_2, s_1, 1)
F_1 = Intersect(d_2, t_1, 1)
G_1 = Intersect(e_2, t_1, 1)
H_1 = Intersect(e_2, a_1, 1)
I_1 = Intersect(f_2, a_1, 1)
J_1 = Intersect(f_2, b_1)
g_2 = Segment(C_1, F_1)
h_2 = Segment(J_1, G_1)
i_2 = Segment(H_1, E_1)
j_2 = Segment(D_1, I_1)
q1, k_2, l_2, m_2, n_2 = Polygon(E_1, C_1, I_1, G_1)
K_1 = Intersect(h_2, j_2)
L_1 = Intersect(h_2, i_2)
q2, p_2, q_2, s_2, t_2 = Polygon(D_1, K_1, L_1, E_1)
M_1 = Intersect(g_2, j_2)
q3, a_2, b_2, f_3, g_3 = Polygon(C_1, M_1, I_1, B_1)
hulknurk5, h_3, i_3, j_3, k_3, l_3 = Polygon(K_1, I_1, G_1, E_1, L_1)
N_1 = Intersect(j_2, k_2)
t1, m_3, n_3, p_3 = Polygon(N_1, M_1, C_1)
q_3 = Segment(K_1, G_1)
O_1 = Intersect(g_2, i_2)
r_3 = Segment(C_1, O_1)
P_1 = (-4.475, -8.47)
Q_1 = (3.04, -8.44)
S_1 = (17.965, -1.03)
T_1 = (28.4, -1.02)
U_1 = (18.225, -9.65)
V_1 = (28.36, -9.66)
W_1 = (5.865, -9.61)
Z_1 = (13.4, -9.56)