jugoslav

Pentagon ABCDE is inscribed in a circle with a center at O and diameter AB. Find the measure of angle AED + angle DCB in degrees.

∠AED + ∠DCB = 270 degrees

AB = AD = 1.632993161

DC = EF = 2*AB = 3.265986322

DE = CF = FG = 2.449489743

[CDEF] = 3.265986322 * 2.449489743 = 8

Diagram not to scale

Let M and N be the midpoints on CD and AB.

MP = 8 / 2 = 4

PN = 4 * AN = 10

Height  H = 14

[ABCD] = (AB + CD)/2 * H = 49 square units

A square sheet of paper ABCD is folded such that B falls on the midpoint M of CD. In what ratio does the crease divide BC?

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BC = 4           y = 4 - x          z = 2

Using Omi's diagram we have:

x² = y² - z²              x = sqrt{ (4 - x)² - 2²}             x = 1.5

Ratio  y : x = 5 : 3

Good job, Guest! I got the same answer: V = 96π cm³

The correct answer is A: BD < AB

Arc RG = 100º      ∠GFR = 1/2 * 100 = 50º

Arc FQ = 28º         ∠FGQ = 1/2 * 28 = 14º

∠FPG = 180 - (50 + 14) = 116º

∠RPG = 180 - ∠FPG = 64º

or

∠RPG = (arc GR + arc FQ) / 2 

∠RPG = (100º + 28º) / 2 = 64º

Consider a square ABCD with side length 2. Let E be the midpoint of AB, F the midpoint of BC, and P and Q the points at which line segment AF intersects DE and DB, respectively.  What is the area of EBQP?

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I'll borrow Phill's diagram.

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Area of Δ ABD = 2

Since DE is a median of Δ ABD, the area of Δ BED = 1

∠ADE = tan^-1(1/2)

DP = cos^-1(∠ADE) * 2

∠EDB = 45 - ∠ADE

PQ = tan(∠EDB) * DP

Area of Δ DPQ = 1/2 (DP * PQ)

[EBQP] = [EDB] - [DPQ] = 7/15     or     0.466666667 square units

 In the diagram, triangle BDF and triangle ECF have the same area. If DB = 2, BA = 3, and AE = 4, find the length of line EC.

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BDCE is a trapezoid.     DC || BE

Triangles ABE and ADC are similar.

AB / AE = AD / AC

3 / 4 = 5 / AC          AC = 20 / 3

EC = 20 / 3 - 4             EC = 2 ²/₃   or  8 / 3    or       2.666666667

In trapezoid ABCD, line AB is parallel to line CD, and AB < CD. The base CD has a length of 8. The legs AD and BC have lengths of 6, and the diagonal BD has a length of 9. Find the area of the trapezoid.

A = 40.06644547