Question

OM is height of AOB, and ON is height of DOC. OM=ON (point O is on line AC). AM=4.2cm, DN=2.6cm. find CD.

(see the picture).
please explain
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Answer 1

• 6.8 cm

Explanation:

As per picture we have:

• OM = ON - given
• ∠AMO≅∠CN) - given
• ∠AOM≅∠CON - vertical angles

This gives us:

• ΔAMO ≅ ΔCON

Two sides and included angle congruent - ASA postulate

As per above congruency we have:

• NC = AM - corresponding sides of congruent triangles

Finding CD:

• CD = DN + NC
• CD = 2.6 + 4.2 = 6.8 cm

Answer 2

`6.8\ cm`

Explanation:

`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\\\\`

`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,\\`

`Lastly,\\We\ are\ given\ that,\\MO=ON`

`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.`

`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`