A-cute Triangle

Geometry Level 4

Let A B C ABC be an acute angled triangle and C D CD be the altitude through C . C. If A B = 8 AB=8 and C D = 6 CD=6 , find the distance between the midpoints of A D AD and B C . BC.


The answer is 5.

Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Ayush G Rai
May 24, 2016

Let E E and F F be the mid-points of A D AD and B C BC respectively.Draw F G FG perpendicular to A B . AB. Then G G is the mid-point of D B . DB. Therefore
E G = E D + D G = 1 2 ( A D + D B ) = 1 2 A B = 4. EG=ED+DG=\frac{1}{2}(AD+DB)=\frac{1}{2}AB=4.
Also F G = 1 2 C D = 3. FG=\frac{1}{2}CD=3. Now from the right angled △ E G F , \triangle EGF,
E F 2 = E G 2 + G F 2 = 4 2 + 3 2 = 5 2 . EF^2=EG^2+GF^2=4^2+3^2=5^2.
and hence E F = 5 . EF=\boxed5.


3 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution (+1)

Ashish Menon - 5 years ago

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