Gegeven is driehoek \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}D = 3 \text{,}\) \(B\kern{-.8pt}D = 1\) en \(B\kern{-.8pt}C = 8 \text{.}\)
3p
Bereken \(D\kern{-.8pt}E \text{.}\)
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\(\triangle A\kern{-.8pt}D\kern{-.8pt}E ∼ \triangle A\kern{-.8pt}B\kern{-.8pt}C\) geeft \({A\kern{-.8pt}D \over A\kern{-.8pt}B} = {D\kern{-.8pt}E \over B\kern{-.8pt}C} = {A\kern{-.8pt}E \over A\kern{-.8pt}C}\)
1p
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\({3 \over 4} = {D\kern{-.8pt}E \over 8} = {A\kern{-.8pt}E \over A\kern{-.8pt}C}\)
1p
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[Kruislings vermenigvuldigen geeft]
\(D\kern{-.8pt}E = {3 ⋅ 8 \over 4} = 6\)
1p
Gegeven is driehoek \(A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}D = 2 \text{,}\) \(B\kern{-.8pt}D = 6 \text{,}\) \(A\kern{-.8pt}C = 7\) en \(B\kern{-.8pt}E = 4 \text{.}\)
3p
Bereken \(D\kern{-.8pt}E \text{.}\)
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\(\triangle A\kern{-.8pt}B\kern{-.8pt}C ∼ \triangle E\kern{-.8pt}B\kern{-.8pt}D\) geeft \({A\kern{-.8pt}B \over B\kern{-.8pt}E} = {B\kern{-.8pt}C \over B\kern{-.8pt}D} = {A\kern{-.8pt}C \over D\kern{-.8pt}E}\)
1p
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\({8 \over 4} = {7 \over D\kern{-.8pt}E}\)
1p
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[Kruislings vermenigvuldigen geeft]
\(D\kern{-.8pt}E = {4 ⋅ 7 \over 8} = 3\frac{1}{2}\)
1p
Gegeven is rechthoek \(A\kern{-.8pt}B\kern{-.8pt}C\kern{-.8pt}D\) met \(A\kern{-.8pt}B = 5 \text{,}\) \(A\kern{-.8pt}D = 10\) en \(C\kern{-.8pt}E = 7 \text{.}\)
4p
Bereken \(B\kern{-.8pt}F \text{.}\)
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\(B\kern{-.8pt}E = B\kern{-.8pt}C - C\kern{-.8pt}E = 10 - 7 = 3 \text{.}\)
1p
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\(\triangle C\kern{-.8pt}D\kern{-.8pt}E ∼ \triangle B\kern{-.8pt}F\kern{-.8pt}E\) geeft \({C\kern{-.8pt}D \over B\kern{-.8pt}F} = {C\kern{-.8pt}E \over B\kern{-.8pt}E} = {D\kern{-.8pt}E \over F\kern{-.8pt}E}\)
1p
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\({5 \over B\kern{-.8pt}F} = {7 \over 3} = {D\kern{-.8pt}E \over F\kern{-.8pt}E}\)
1p
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[Kruislings vermenigvuldigen geeft]
\(B\kern{-.8pt}F = {5 ⋅ 3 \over 7} = 2\frac{1}{7}\)
1p
Gegeven is rechthoek \(A\kern{-.8pt}B\kern{-.8pt}C\kern{-.8pt}D\) met \(A\kern{-.8pt}B = 5 \text{,}\) \(A\kern{-.8pt}D = 6\) en \(B\kern{-.8pt}F = 4 \text{.}\)
4p
Bereken \(C\kern{-.8pt}E \text{.}\)
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\(\triangle B\kern{-.8pt}F\kern{-.8pt}E ∼ \triangle A\kern{-.8pt}F\kern{-.8pt}D\) geeft \({B\kern{-.8pt}F \over A\kern{-.8pt}F} = {F\kern{-.8pt}E \over F\kern{-.8pt}D} = {B\kern{-.8pt}E \over A\kern{-.8pt}D}\)
1p
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\({4 \over 9} = {F\kern{-.8pt}E \over F\kern{-.8pt}D} = {B\kern{-.8pt}E \over 6}\)
1p
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[Kruislings vermenigvuldigen geeft]
\(B\kern{-.8pt}E = {4 ⋅ 6 \over 9} = 2\frac{2}{3}\)
1p
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\(C\kern{-.8pt}E = B\kern{-.8pt}C - B\kern{-.8pt}E = 6 - 2\frac{2}{3} = 3\frac{1}{3} \text{.}\)
1p
Gegeven is driehoek \(\triangle A\kern{-.8pt}B\kern{-.8pt}C\) met \(B\kern{-.8pt}C = 11 \text{,}\) \(B\kern{-.8pt}D = 8\) en \(D\kern{-.8pt}E = 3 \text{.}\)
4p
Bereken \(A\kern{-.8pt}D \text{.}\)
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\(\triangle D\kern{-.8pt}A\kern{-.8pt}E ∼ \triangle B\kern{-.8pt}A\kern{-.8pt}C\) geeft \({A\kern{-.8pt}D \over A\kern{-.8pt}B} = {A\kern{-.8pt}E \over A\kern{-.8pt}C} = {D\kern{-.8pt}E \over B\kern{-.8pt}C}\)
1p
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Neem \(A\kern{-.8pt}D = x \text{,}\) dan geldt \(A\kern{-.8pt}B = x + 8\) en dus
\({x \over x + 8} = {A\kern{-.8pt}E \over B\kern{-.8pt}C} = {3 \over 11}\)
1p
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[Kruislings vermenigvuldigen geeft]
\(11 x = 3 (x + 8)\)
1p
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\(11 x = 3 x + 24\)
\(8 x = 24\)
\(x = {24 \over 8} = 3 \text{,}\) dus \(A\kern{-.8pt}D = 3 \text{.}\)
1p
Gegeven is driehoek \(A\kern{-.8pt}B\kern{-.8pt}C\) met \(A\kern{-.8pt}D = 10 \text{,}\) \(B\kern{-.8pt}D = 6 \text{,}\) \(A\kern{-.8pt}C = 8\) en \(C\kern{-.8pt}E = 4 \text{.}\)
5p
Bereken \(B\kern{-.8pt}E \text{.}\)
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\(\triangle A\kern{-.8pt}B\kern{-.8pt}C ∼ \triangle E\kern{-.8pt}B\kern{-.8pt}D\) geeft \({A\kern{-.8pt}B \over B\kern{-.8pt}E} = {B\kern{-.8pt}C \over B\kern{-.8pt}D} = {A\kern{-.8pt}C \over D\kern{-.8pt}E}\)
1p
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Neem \(B\kern{-.8pt}E = x \text{,}\) dan geldt \(B\kern{-.8pt}C = x + 4\) en dus
\({16 \over x} = {x + 4 \over 6}\)
1p
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[Kruislings vermenigvuldigen geeft]
\(x (x + 4) = 96\)
1p
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\(x^{2} + 4 x - 96 = 0\)
\((x - 8) (x + 12) = 0\)
\(x = 8 ∨ x = -12\)
1p
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[Een lengte is altijd positief, dus] \(B\kern{-.8pt}E = 8 \text{.}\)
1p