Gegeven is de volgende figuur met ADN een gelijkzijdige driehoek en
MC = MN = MB.
a. Leg uit waarom er een cirkel door de punten B, C en
N gaat en bepaal de lengte van de straal.
b. Bereken exact de lengte van AB als je weet dat MN =
10.
Gegeven is nu alleen dat de omtrek van ABCD gelijk is
aan 100.
c. Bereken exact de lengte van AD en schrijf deze in de vorm 𝐴𝐴𝐴𝐴 = 𝑎𝑎 + 𝑏𝑏√𝑐𝑐.
, Hoofdstuk 14 Meetkunde Toepassen (V6 Wis B) Pagina 2 van 15
OPLOSSING 1
a. Als je een cirkel tekent met middelpunt M en straat MC gaat deze door N en B omdat MC
= MN = MB.
b. Noem P het punt halverwege AD. Je krijgt dan een 30-60-90 driehoek :
N
D
P
Er geldt 𝐴𝐴𝐴𝐴 = 𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀 = 10
Dus 𝑃𝑃𝑃𝑃 = 10√3
Dus 𝐴𝐴𝐴𝐴 = 𝑃𝑃𝑃𝑃 + 𝑀𝑀𝑀𝑀 = 10√3 + 10
c. Nu geldt dat 𝐴𝐴𝐴𝐴 = 𝑥𝑥.
Dus 𝐴𝐴𝐴𝐴 = 𝑃𝑃𝑃𝑃 + 𝑀𝑀𝑀𝑀 = 𝑥𝑥√3 + 𝑥𝑥 en 𝐴𝐴𝐴𝐴 = 2𝑃𝑃𝑃𝑃 = 2𝑥𝑥
50 50
Omdat AD = 2AP is dus 𝐴𝐴𝐴𝐴 = 2 �− √3 − 25� = − √3 + 50
6 3
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