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Let $AC$ be a diameter of a circle $\omega$ of radius $1$, and let $D$ be the point on $AC$ such that $CD = \frac{1}{5}$. Let $B$ be the point on $\omega$ such that $DB$ is perpendicular to $AC$, and let $E$ be the midpoint of $DB$. Compute length $AE$.

 Jan 14, 2024
 #1
avatar+128732 
+1

OC  =1

OD  = OC - CD =  1 -1/5 = 4/5 

BD = sqrt [ 1 - (4/5)^2 ] =  sqrt [ 9/25 ]  = 3/5 

ED = (3/5) (1/2) =  3/10

AD = 1 + 4/5 = 1 + 8/10  = 18/10

 

Triangle ADE is  right with angle ADE =  90......so....using the P Theorem .....

 

AE = sqrt [ AD^2 +ED^2 ]  = sqrt [ (18/10)^2  + (3/10)^2 ]  = sqrt [ 333 ] / 10 ≈  1.825

 

cool cool cool

 Jan 14, 2024
edited by CPhill  Jan 14, 2024
edited by CPhill  Jan 14, 2024
 #2
avatar+289 
+1

Given:

 

AC is the diameter of a circle \(\omega\)

circle \(\omega\) has radius 1

D is a point on AC such that CD = 1/5

B is a point of circle \(\omega\) such that DB is perpendicular to AC

E is the midpoint of DB

 

Question:

 

AE = ?

 

I will assume that B is one the circumference of the circle, or else, it is not possible to find AE.

 

Using https://jspaint.app/#local:cf1101d801c0e, we can draw this figure:

 

 

We can calculate length AE by doing AD^2 + ED^2. 

 

We know that AD is 2 - 1/5 = 1 4/5, and squaring that gives us 3.24, so we get:

 

3.24 + ED^2 = AE^2

 

To calculate length ED, we first need to find length BD, which can be found by drawing a line OB, and then using the Pythagorean Theorem:

 

OD^2 + BD^2 = OB^2

 

We know that OD is 4/5 and OB is 1, so we can solve for BD:

 

0.8^2 + BD^2 = 1

 

0.64 + BD^2 = 1

 

BD^2 = 0.36

 

BD = square root of 0.36 or sqrt 36/100 = 6/10 = 0.6

 

BD = 0.6, so the midpoint divides BD into 2 equal segments, which as equal to 0.6/2 = 0.3, so ED = 0.3.

 

Using the Pythagorean Theorem to find AE, we get:

 

AD^2 + ED^2 = AE^2

 

1.8^2 + 0.3^2 = AE^2

 

3.24 + 0.09 + AE^2

 

AE^2 = 3.33

 

AE = square root of 3.33 or approximately 1.825.

 

The answer is \(\sqrt{3.33}\) or 1.825

 

Answer: \(\sqrt{3.33}\) or \(1.825\)

 

Okay we got the same answer.

 Jan 14, 2024
edited by DS2011  Jan 14, 2024

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