# Trapezoid Problem

• Dec 3rd 2006, 01:46 PM
Kolbeck10
Trapezoid Problem
Given trapezoid ABCD with diagonals intersecting in X, find three pairs of triangles with equal areas.

http://img106.imageshack.us/img106/3...w834817ry3.gif

How would one go about doing and proving that?
• Dec 3rd 2006, 02:14 PM
CaptainBlack
Quote:

Originally Posted by Kolbeck10
Given trapezoid ABCD with diagonals intersecting in X, find three pairs of triangles with equal areas.

http://img106.imageshack.us/img106/3...w834817ry3.gif

How would one go about doing and proving that?

ABD and ABC have equal base (AB) and altitude and hence equal area.

CDB and CDA for the same reason have equal area.

AXD and BXC, as ABD and ABC have the same area and:

area(ABD)=area(AXD)+area(AXB)

area(ABC)=area(BXC)+area(AXB).

RonL
• Dec 3rd 2006, 02:19 PM
Kolbeck10
Quote:

Originally Posted by CaptainBlack
ABD and ABC have equal base (AB) and altitude and hence equal area.

CDB and CDA for the same reason have equal area.

AXD and BXC, as ABD and ABC have the same area and:

area(ABD)=area(AXD)+area(AXB)

area(ABC)=area(BXC)+area(AXB).

RonL

Is this dependent on the fact that it's an isosceles trapezoid, or would it be that way for one where the two non-parallel sides aren't the same?
• Dec 3rd 2006, 03:02 PM
CaptainBlack
Quote:

Originally Posted by Kolbeck10
Is this dependent on the fact that it's an isosceles trapezoid, or would it be that way for one where the two non-parallel sides aren't the same?

No, I tried to ignore that aspect of the diagram.

RonL
• Dec 3rd 2006, 03:19 PM
Kolbeck10
Quote:

Originally Posted by CaptainBlack
No, I tried to ignore that aspect of the diagram.

RonL

Alright, perfect, thanks for the help. :)