MathDB

Set 4
4.1 Quadrilateral

ABCD

(with

A, B, C

not collinear and

A, D, C

not collinear) has

AB = 4

BC = 7

CD = 10

, and

DA = 5

. Compute the number of possible integer lengths

AC

. https://cdn.artofproblemsolving.com/attachments/1/6/4f43873a64bc00a0e6173002ccd80e8f1529a9.png
4.2 Let

T

be the answer from the previous part.

2T

congruent isosceles triangles with base length

b

and leg length

\ell

are arranged to form a parallelogram as shown below (not necessarily the correct number of triangles). If the total length of all drawn line segments (not double counting overlapping sides) is exactly three times the perimeter of the parallelogram, find

\frac{\ell}{b}

. https://cdn.artofproblemsolving.com/attachments/5/c/744f503ed822bc43acafe2633e6108022f2c88.png
4.3 Let

T

be the answer from the previous part. Rectangle

R

has length

T

times its width.

R

is inscribed in a square

S

such that the diagonals of

S

are parallel to the sides of

R

. What proportion of the area of

S

is contained within

R

? https://cdn.artofproblemsolving.com/attachments/a/1/0928dd1ffbeb4d7dee9b697fdb7696cc70c03d.png
PS. You should use hide for answers.

Problem Statement