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Isosceles Trapezoid

Date: 6/23/96 at 1:21:41
From: Anonymous
Subject: Isosceles Trapezoid

The bases AB and CD of an isosceles trapezoid ABCD are 12 units apart. 
AB = 10 and CD = 8. Point R is on the axis of symmetry of the 
trapezoid so that the angle CRB measures 90 degrees. Find the possible 
distances from R to AB.

I don't even know where to begin.  Please help!

Date: 6/23/96 at 8:12:30
From: Doctor Anthony
Subject: Re: Isosceles Trapezoid

Draw the following diagram and refer to it as I go through the work.

Draw the isosceles trapezium ABCD and let E be midpoint of AB and     
F the midpoint of CD. Draw in the lines CR and BR to meet at right 
angles on the line EF. 

We now let RE = x so that RF = 12-x and we consider the two right-
angled triangles BRE and RCF. These triangles are similar because they 
are both right-angled and angle BRE = angle RCF (both are 
complementary to angle CRF). 

We can equate ratios of corresponding sides, so BE/RE = RF/CF. 
Filling in the lengths we get:

5/x = (12-x)/4

20 = x(12-x) 

20 = 12x - x^2

x^2 - 12x + 20 = 0

(x-2)(x-10) = 0    so x = 2  or x = 10

So point R can be 2 units or 10 units from point E.

-Doctor Anthony,  The Math Forum
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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