A line, which goes through the point of intersection of the diagonals of a trapezoid, divides one of the bases into two segments. The ratio of the length of these segments is m:n. What is the ratio of the length of the segments of the other base?

The diagonals (AC, BD) of a trapezoid (ABCD) divide the figure into triangles. Those triangles having parallel bases are similar (ABP, CDP). A line through the common point of those triangles (P, the intersection point of the diagonals) will further divide those similar triangles into similar triangles.

The ratio of the base segments will be the same (AX:XB = CY:YD). Since one triangle is "upside down" with respect to the other one, if the segment lengths are considered from the same side of the trapezoid, they are in the ratio m:n (=AX:XB) on one base and n:m (=DY:YC) on the other base.

MrRoyal MrRoyal · Answer 2 · 2021-11-29T13:53:31+01:00

The point of intersection of lines is the point where the lines meet.

The ratio of the length of the segments of the other base is CY : YD

The ratio is given as

m : n

From the complete question, we have the following ratio

m : n = AX : XB

By comparison:

m = AX

n = XB

Also from the complete question, we have:

AX = CY

XB = YD

This means that:

m = AX = CY

n = XB = YD

So, we have the following ratios

m : n = AX : XB

m : n = CY : YD

Hence, the other ratio is CY : YD