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Triangle ABC is acute. Point D lies on AC so that line BD is perpendicular to line AC, and point E lies on AB such that line CE is perpendicular to line AB. The intersection of segments line CE and line BD is P. Find the value of AE if CP=10, PE=15, and EB=30.

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ΔACE ~ ΔBPE

AE / CE = PE / EB               AE = (25 * 15) / 30

AE = 12.5

                   B

                   l

                   l

                   l

                   l

           30    l

                   l

                   l

        E         l

             15  l

                   P

                   l      10

A                D              C

BP =sqrt (30^2 +15^2)   = sqrt (1125)= 15sqrt (5)

sin BPE =  30 / (15 sqrt (5))  = 2/sqrt (5)  = sin CPD

sin PCD  =  sqrt ( 1 - (2/sqrt (5))^2)  =  sqrt ( 1 -4/5)   =  1/sqrt (5)

PD / sin PCD =  PC /sin 90

PD =  10 / sqrt (5) =   2sqrt (5)

BD =  17sqrt (5)

angle  BPE =  angle BAD

BD / sin BAD = BA / sin  BDA

17sqrt (5) /( 2 /sqrt (5))  =  (30 +AE) / sin 90

(17 * 5)/2  = 30 + AE

85/2   =  30 + AE

42.5  = 30  + AE

AE =  42.5  - 30   =  12.5

if we convert each number to make the denominator 2^10, we get 2^9+2^8...1(all the numerators).

If we add powers of 2 starting from one, it is always one less than the nest power of 2

1+2 is 1 less than 4, 1+2+4 is 1 less than 8, and so on. 

So it would be (2^10-1)/2^10, or 1023/1024.

Use the formula for the nth term of a GS:

Nth term =First term x Common ratio^(Term number - 1)

1/4           = 2                 x     R^(2 - 1)

1/4           = 2                 x      R^1

Divide both sides by 2

(1/4) / 2  = R

1 / 8         = R, or:

R = 1 / 8 - Common ratio

5th term =First term x Common ratio^(term number - 1)

5th term = 2                x    (1 /8)^(5 - 1)

5th term = 2                x    (1/8)^4

5th term = 2                x      1 / 4096

5th term = 2 /4096 = 1 / 2048

-6/6 is -1, so it keeps on multiplying by negative one. Looking at the pattern, we see that the number six is in the odd positions. Since 205 is odd, the answer is 6.

The answer from the above link is incorrect!!!

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In triangle ABC, AB=AC and D is a point on line AC so that line BD bisects angle ABC. If BD=BC, what is the measure, in degrees, of angle A?

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AB = AC        BD = BC          AD = BD = BC  

The sum of the interior angles of each triangle is 5x

 5x = 180        x = 36          ∠A = 36º

Thx CPhill also their is another way this is how I did it.

The first term is 3, and the ratio between terms is (9/2)/3=3/2 . Therefore, the eighth term of the sequence is \(3\cdot(3/2)^{8-1} = 3^8/2^7 = \boxed{6561/128}\).

The common  ratio is    (9/2)  / 3  =   9/6  = 3/2

The eighth term is

3 ( 3/2)^(8- 1)  =   3 (3/2)^7    =

3 ( 2187 / 128)  =

6561/ 128

At each stop, the number of students on the bus is cut in half. Therefore, after 3 stops, the number of students on the bus is \(48(\frac12)^3 = \frac{48}8 = \boxed{6}\) .

48 (1/2)^3  =

48 * (1/8)  =  

6   .....just as    4M found   !!!

Answer:

2048/4=512

512/4=128

128/4=32

32/4=8

So x=32 ; y=8

x+y=40

Let A =(0,0)  B =(25,0)  C =(25,25) D =(0,25)

APB  is a 7-24-25 right triangle 

Its area is   (1/2) ( 7) (24) = 84

Also...its  area  =   (1/2)  AB   *  height

84 =  (1/2) 25  * height

168/25  = height =  y coordinate of P  = 6.72

And using the  equation  of a circle  x^2  + y^2 = 24^2...we can find the x coordinate as

x^2 +  (6.72)^2  =  24^2

x=  sqrt  [  24^2 - ( 6.72)^2  ]  =  23.04

So DP^2   =

( 23.04)^2  +  ( 25 - 6.72)^2   =   865

Also.....note something interesting.... 

AP^2  + CP^2  =  BP^2  + DP^2

24^2  + CP^2   = 7^2  + 865

576  + CP^2   = 914

CP^2 = 914 - 576   =   338

Great job, Pangolin! I was thinking about this problem but I couldn't come up with non-trig solutions.

243/81 is 3, so

81/3=27, and

27/3, is 9.

243, 81, 27, 9 , 3.

X+y=27+9=36.

First stop - 48/2 =24

Second stop - 24/2=12

Final stop - 12/2=6

6 students remain.

Each term gets smaller by dividing by 3, so term 1 is, term 2 is \({9}/3^1\), and so on. Continuing the pattern, number 10 would be \(9/3^9 or 1/3^7 or 1/2187\)

Or, would it just be 24 * 25/300?

I didn't use trig- I'll explain.

The incenter coordinates are $\left(\frac{1}{2} (-a + b + c), \frac{\sqrt{-(a - b - c) (a + b - c) (a - b + c) (a + b + c)}}{{2 (a + b + c)}} \right)$ where a,b,and c are the side lengths of a triangle. Then we have that the incenter's coordinates, if we let a=8 and b = c, are $\left(\frac{1}{2} (-8 + b + b), \frac{\sqrt{-(8 - b - b) (8 + b - b) (8 - b + b) (8 + b + b)}}{2 (8 + b + b)}\right)$ which simplifies to $\left(\frac{1}{2} (2 b - 8), 2 \frac{\sqrt{(2 b - 8) (2 b + 8)}}{ b + 4}\right).$ If we set the right side of the equation equal to 3, we can easily cross multiply and get $b = \frac{100}{7}.$ Thus, AB = $\frac{100}{7},$ and we may want to use heron's formula, but for a more convenient solution we can say that A = (0,a), and using the distance formula with B = (-4,0), we have $\sqrt{a^2-16} = 100/7,$ so $a = \frac{96}{7},$ and the height is $\frac{96}{7}$

See the following  :

Angle EAB  =22.5°

Angle AEB = 67.5°

Using the Law of Sines

BE /sin22.5 =  1/sin AEB

BE =   sin 22.5 / sin 67.5

BE  = sin22.5/ cos22.5

BE  = tan 22.5  =    (1 -cos 45)/ sin 45  =   (1 - 1/sqrt (2)) sqrt (2)  =   sqrt (2)  - 1

So the area of  triangle AEB =  (1/2)  ( AB) ( BE)  = (1/2) (1) (sqrt (2) - 1)

And this  = area of triangle ADF

So  [ AEF ]  =  [ area of square ]  - [ CEF] - 2 [ AEB]  =

1  -  (1/8)  -  ( (sqrt (2)  - 1  )  =

2 - 1/8  - sqrt (2)  =

15/8 -  sqrt (2) ≈  .461  units^2

https://web2.0calc.com/questions/geometry-help_68

I meant 2√7

In a certain isosceles right triangle, the altitude to the hypotenuse has length 4. What is the area of the triangle?

Area = 4² 

See the image here :

The  perimeter of the  triangle is  ( 6 + 8)  + (6 + 7)  + ( 8 + 7)  =  14 + 13 + 15  =  42

The area  =  (1/2) radius of incircle  * perimeter  = 

(1/2) (4) (42)  = 

2 *  42  =  

84    .....just as the  guest found  !!!!

Yes, it is 84!

Find angles A, B, and C

Find side lengths

Use Heron's formula to calculate the area.

The answer is 84 don't have any time for an explanation today. Hope this is correct.