1. (17 pts) In general, there's no spectacular relationship between the side lengths of a quadrilateral and the lengths of its diagonals. BUT, if the quadrilateral happens to have all four vertices lying on the same circle, there is something special that happens. And that's the point of this problem: Given quadrilateral ABCD, where A, B, C, and D all lie on the same circle, prove that AB x DC AD × BC = AC × BD To make your life easier, just use the following steps. First, construct AX so that ZDAXZBAP. DX AD (a) Show that CB AC XB AB (b) Show that = DC AC D Χ B P C (c) Use parts (a) and (b) to evaluate DX + XB. Finish the proof!

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter4: Quadrilaterals
Section4.3: The Rectangle, Square, And Rhombus
Problem 5E: If the diagonals of a parallelogram are perpendicular and congruent, what can you conclude regarding...
Question
1. (17 pts) In general, there's no spectacular relationship between the side lengths of a
quadrilateral and the lengths of its diagonals. BUT, if the quadrilateral happens to
have all four vertices lying on the same circle, there is something special that happens.
And that's the point of this problem:
Given quadrilateral ABCD, where A, B, C, and D all lie on the same circle, prove
that
AB x DC AD × BC = AC × BD
To make your life easier, just use the following steps. First, construct AX so that
ZDAXZBAP.
DX
AD
(a) Show that
CB
AC
XB
AB
(b) Show that
=
DC
AC
D
Χ
B
P
C
(c) Use parts (a) and (b) to evaluate DX + XB. Finish the proof!
Transcribed Image Text:1. (17 pts) In general, there's no spectacular relationship between the side lengths of a quadrilateral and the lengths of its diagonals. BUT, if the quadrilateral happens to have all four vertices lying on the same circle, there is something special that happens. And that's the point of this problem: Given quadrilateral ABCD, where A, B, C, and D all lie on the same circle, prove that AB x DC AD × BC = AC × BD To make your life easier, just use the following steps. First, construct AX so that ZDAXZBAP. DX AD (a) Show that CB AC XB AB (b) Show that = DC AC D Χ B P C (c) Use parts (a) and (b) to evaluate DX + XB. Finish the proof!
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