Tough plane geo question

bonkysleuth SG

In the triangle ABC, BF and CE are perpendicular to AC and AB respectively. D and G are the midpoints of EF and BC respectively. Prove that GD is perpendicular to EF.

Been doing plane geo questions since morning. argh, driving me crazy. Can someone tell me how to score for such questions? TYS answers are as good as not giving at all. The more questions on plane geo I do, the more insecure and less confident I become... having a maths exam tomorrow and my teacher said plane geo question is worth 9 marks.

thanks everyone

Edited by mod for Typos.

skythewood SG

I try to find the triangle ABD. I failed.

yiha093 SG

use triangle properties ?

eagle SG

ok this question took me about 5+ mins to solve, quite tough... :(

dot a perpendicular line from G to BE and label it as H
dot a perpendicular line from G to FC and label it as I

Note that by similar triangles,

consider triangles HBG and EBC
because BG = ½BC,
EH = ½BE
HG = ½EC

consider triangles GIC and BFC
because BG = ½BC,
FI = ½FC
GI = ½BF

Thus,

EG² = EH² + HG²
= ¼BE² + ¼EC²
= ¼BC²

GF² = GI² + FI²
= ¼BF² + ¼CF²
= ¼BC²
= EG²

Thus, GF = EG

Since GF = EG, and ED = DF, then GEF is an isosceles triangle where GD is perpendicular to EF

wee_ws SG

Hi eagle,

Clever!

After drawing the extra lines (we call them auxiliary lines), we can use the intercept theorem directly. Then it is followed by the application of Pythagoras' theorem.

Thanks for sharing!

Cheers,
Wen Shih

eagle SG

Thanks Wen Shih

I have drawn out the answer and placed it on examworld

Enjoy :D