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Suppose that triangle ABC is a right triangle with the right angle at C. Let line segment CD be the perpendicular from point C to the hypotenuse line segment AB. Show that the ratio of the areas of triangles ADC and DCB is the same as the ratio AD: DB

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4 votes

Explanation:

Given:

Let triangle ACD is aright angle triangle with right angle at C. A line perpendicular to AB join C.

Therefore we can say that line segment CD divides angle at C into two equal angles.

So in ΔACD and ΔCDB

∠ ACD = ∠DCB

and ∠ADC = ∠BDC = 90°

and CD =CD

∴ we can say that ΔACD and ΔCDB are similar triangles.

∴ Area of ΔACD =
(1)/(2)* base* height

=
(1)/(2)* AD* CD

Area of ΔCDB =
(1)/(2)* base* height

=
(1)/(2)* DB* CD

Therefore ratio of the areas of ΔACD and ΔCDB is

i.e.
(\Delta ACD)/(\Delta CDB) =
((1)/(2)* AD* CD)/((1)/(2)* DB* CD)

=
(AD)/(DB)

∴ Area of the ratio of
(\Delta ACD)/(\Delta CDB) =
(AD)/(DB)

Hence proved

User Mark Karavan
by
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