CPhill

Sine of an angle  = the ratio of the opposite side to the hypotenuse

So....for angle R

Opposite side  = 5

Hypotenuse = 13

So

sin R   = 5 / 13

Here's the first one

Actually....angles  A and C are equal

And angles  E and C are equal

So....by AA conguency.....triange ACB is similar to triangle ECD

So....angle  BAC = angle DEC...and these are opposite imterior angles 

And angle CBA = angle EDC...and these are oppositie interior angles

And AE and BD are transversals cutting AB and DE

But...when transversals cutting two line segments make opposite interior angles equal....the segments are parallel

So   AB ll DE

Heck.....tertre beat me to it.....LOL!!!

So.....let b be the base.....and we have

(1b + 4)  + ( 2b +4)    = 4b + 1      simplify

3b + 8   = 4b + 1         subtract 3b, 1 from both sides

b = 7

Check

14₇ + 24₇ =  41₇

In base 10

11 + 18  = 29     ⇔   True!!!

We know an angle and the adjacent side to this angle

And we are looking for the hypotenuse [ the side opposite the right angle ]

The cosine  = the ratio of the adjacent side to the hypotenuse....so

cos (20°) =   54 / x  ⇒ "C"

We have a right triangle, NSS....we know an angle and an opposite side to the angle

And we are looking for the side adjacent to the angle, x

[The adjacent side is the side next to the angle.....but not the hypotenuse....[ the side opposite the right angle]

The tangent of an angle = ratio of the its opposite side to its adjacent side....so

tan (17°) =    200 / x     multiply both sides by x

x * tan (17”)  = 200      divide bothsides by tan (17°)

x =   200 / tan (17°)  = 654 ft

Hope that helps, NSS....

3, 3, 3, 3, 3, 3, 3, 3, 2

We take this this in reverse order to find the continued fraction....it is actually not as hard as it seems...just a  (rather tedious) iterative process

So....We work from right to left

Take the last 2, invert it ⇒ 1/2....add it to the next-to-the-last 3   = 3 + 1/2 = 7/2 

Invert this = 2/7 and add it to the next 3 to the left = 3 + 2/7 = 23/7

Invert this =  7/23 and add it to the next 3 to the left   = 3 + 7/23 = 76/23

Invert this and add it to the next  3 on the left = 23/76 + 3 = 251/76

Invert this and add it to the next 3 on the left = 76/251 + 3  = 829/251

Invert this and add it to the next 3 on the left = 251 / 829 + 3 = 2738 / 829

Invert this and add it to the next 3 on the left = 829 / 2738 + 3 = 9043/2738

Invert this and add it to the last 3 on the left = 2738/9043 + 3  = 29867 / 9043  =  a / b

Nice, Heureka....!!!!

Thanks for that answer, Neworleans   !!!!!

If we conect WY.....we have an isosceles right triangle.... WY = 4sqrt (2)  = √32

And angle  WYX   = 45.....so angle WYZ = 135 - 45 = 90

So...triangle WYZ is a right triangle with WZ the hypotenuse and  WY a leg

So....by the Pythagorean Theorem     

a =  √ [ WZ^2 - WY^2 ] = √ [ 9^2 - (√32)^2 ] = √ [ 81 - 32 ]  = √49 =  7