computational in an isosceles trapezoid, inradius given

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

computational in an isosceles trapezoid, inradius given

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

Source: Vyacheslav Yasinsky Geometry Olympiad 2017 X-XI p1 [Ukranie]

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30629 posts

#1 h Sep 22, 2018, 4:00 AM • 1 Y

Y by Adventure10

In the isosceles trapezoid with the area of $28$ , a circle of radius $2$ is inscribed. Find the length of the side of the trapezoid.

This post has been edited 3 times. Last edited by parmenides51, Sep 22, 2018, 4:31 AM
Reason: source edit

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

472 posts

Ari004

#2 h Sep 22, 2018, 7:17 AM • 2 Y

Y by Adventure10, Mango247

Suppose in the isosceles trapezoid $ABCD$ , $AB \parallel CD$ , $I$ is the incenter and $AB=2x,BC=AD=h,CD=2y$ .Let $G$ and $H$ be the projections of $I$ onto $AB$ and $CD$ and WLOG $AB<CD$ .Easy to see that $G,I,H$ are collinear and by symmetry, $G,H$ are the midpoints of $AB,CD$ .Since the area of $ABCD=28$ , $x+y=\frac{28}{4}=7$ .By Pitot's theorem on $ABCD$ ,we have $2(x+y)=2h$ or $x+y=7=h$ .Let $X$ be the projection of $B$ onto $CD$ .Considering the right angled $\bigtriangleup BXC$ ,we have $BX=4,CX=7-2x,BC=7$ ,by Pythagoras theorem, $x=\frac{7-\sqrt{33}}{2}$ and we are done.

This post has been edited 5 times. Last edited by Ari004, Sep 22, 2018, 7:23 AM

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a