Answer:
[tex]6.8\ cm[/tex]
Step-by-step explanation:
[tex]We\ are\ given:\\OM\ is\ altitude\ of\ \triangle AOB\\ON\ is\ altitude\ of\ \triangle DOC\\Now,\\Lets\ only\ focus\ on\ the\ interior\ triangles\ inside\ the\ two\ above\ triangles.\\They\ are:\\\triangle OMA\ and\ \triangle ONC.\\\\We\ observe\ that:\\\angle AMO=90[As\ it\ is\ the\ altitude\ of\ \triangle AMO]\\\angle ONC=90[As\ it\ is\ the\ altitude\ of\ \triangle ONC]\\Hence,\\\angle AMO= \angle ONC=90\\\\[/tex]
[tex]We\ also\ observe\ that,\\\angle MOA\ and\ \angle CON\ are\ vertically\ opposite\ as\ they\ are\ one\ pair\\ of\ opposite\ angles\ formed\ by\ the\ intersection\ of\ Lines\ AC\ and\ MN.\\We\ know\ that,\\"When\ two\ straight\ lines\ intersect,\ the\ pairs\ of\ vertically\ opposite\\ angles\ formed\ are\ equal".\\Hence,\\\angle MOA= \angle CON[/tex]
[tex]Lastly,\\We\ are\ given\ that,\\MO=ON[/tex]
[tex]Now,\\The\ ASA\ Congruence\ Criterion\ states\ that' If\ two\ angles\ and\ the\ includ\\ side\ of\ one\ triangle\ is\ equal\ to\ the\ corresponding\ angles\ and\ the\ includ\\ side\ of\ the\ second\ triangle,\ both\ triangles\ are\ congruent'.\\Hence,\\\triangle AMO \cong \triangle CNO.[/tex]
[tex]Hence,\\AM=NC [Corresponding\ Parts\ of\ Congruent\ Triangles]\\We\ can\ observe\ that,\\CD=DN+NC\\Hence,\\DN=2.6\ cm [Given]\\NC=AM=4.2\ cm [Proven]\\Hence,\\DC=2.6+4.2=6.8\ cm[/tex]
