Question

Secant DB intersects DZ at point D. Find the length of DA. 3x*(3x+8)=x*(x+32)

Answer 1

if two secants intersect outside the circle, the product of the measures of one secant and its external segment equals the product of the measures of the other secant and its external segment.

Answer 2

to interpret that,
we need: total length of each secant.
any idea how to write the length of each secant (using the info they gave) ?

Answer 3

3x*(3x+8)=x*(x+32)

that looks correct

to solve for x (don't forget we really want DA which is 3x)
distribute the 3x on the left side and the x on the right side
do you know how to do that ?

Answer 4

9x+3x(8)=34x ?? I don't think I did that right.

Answer 5

no, but you did try.

distribute means multiply each thing inside the parens
it (maybe?) makes sense for numbers:
say you had 3*(1+2)
if you think of (1+2) as a package with 1 and 2 inside it, and you have 3 packages
it might make sense you have 3 ones and 3 twos all together
in "math" we would write it
3*(1+2) = 3*1 + 3*2
which "says" the same thing if you learn how to read it.

Answer 6

in this problem , which has letters, we can still use the same idea
3x*(3x+8)=x*(x+32)

we do 3x times 3x and 3x times 8 (on the left side)

to do that, we write
3x*3x + 3x*8
and it helps to remember 3x is short for 3 times x or 3*x
you have
3*x*3*x + 3*x*8

when you multipy things, you can switch the order
3*3*x*x + 3*8*x
for example
and you know how to simplify 3*3 (to get 9)
and 3*8 is 24
9*x*x + 24*x

on the other side
x*(x+32)
you distribute the x
x*x + x*32

usually people use the "short-cut" to write x*x as \( x^2\)

we would normally not bother showing the multiply sign, and put numbers in front of letters. in other words
9*x*x + 24*x = x*x + x*32
is usually written as
\[ 9x^2 +24x = x^2 +32x\]

Answer 7

now you use algebra to solve that equation
add -x^2 to both sides
\[ 9x^2 - x^2 + 24x = x^2 - x^2 +32x\]
on the right side you have 1 x^2 and you take away 1 x^2. that leaves no x^2 on that side
on the left side
you have 9 x^2 take away 1 x^2
that leaves 8 x^2 on the left side
\[ 8x^2 +24x= 32x\]

now add -32x to both sides
can you try ?

Answer 8

\[8x ^{2} + 24x -32x = 32x - 32x\]

Answer 9

looks good

Answer 10

\[8x ^{2}-8x = 0\]

Answer 11

yes.
now factor out an x
\[ x(8x-8) = 0\]

when you multiply two things and get zero, that means one or the other is zero
in other words, either
x=0 or 8x-8=0
(we don't care about the x=0 solution)
what do you get for x from
8x-8= 0
?

Answer 12

\[8x = 8\]

Answer 13

x = 1 ?

Answer 14

yes
now Find the length of DA.

Answer 15

DA is 3x
so not too hard to finish this up.

Answer 16

DA is 3

Answer 17

Thank you so much!!