Sure, let's go through this problem step-by-step.
1. Use [tex]\(\triangle CBD\)[/tex] to create an equation for [tex]\(a^2\)[/tex]:
In the triangle [tex]\(\triangle CBD\)[/tex], let [tex]\(h\)[/tex] be the height dropped from point [tex]\(D\)[/tex] to the base [tex]\(BC\)[/tex]. We know:
[tex]\[
h^2 = a^2 - x^2
\][/tex]
This relationship comes from the Pythagorean theorem applied to the right triangle [tex]\(\triangle BHD\)[/tex], where [tex]\(BD = a\)[/tex] is the hypotenuse, [tex]\(BH = x\)[/tex] is one leg, and the height [tex]\(h\)[/tex] is the other leg.
2. Solve this equation for [tex]\(h^2\)[/tex]:
From the relationship above, we already have [tex]\(h^2\)[/tex] isolated:
[tex]\[
h^2 = a^2 - x^2
\][/tex]
3. Replace your expression for [tex]\(h^2\)[/tex] in the equation [tex]\(c^2 = h^2 + b^2 - 2bx + x^2\)[/tex]:
Now, we can substitute [tex]\(h^2\)[/tex] from the equation found in step 2 into the given equation [tex]\(c^2 = h^2 + b^2 - 2bx + x^2\)[/tex]:
[tex]\[
c^2 = (a^2 - x^2) + b^2 - 2bx + x^2
\][/tex]
4. Simplify the equation:
Combine like terms:
[tex]\[
c^2 = a^2 - x^2 + b^2 - 2bx + x^2
\][/tex]
Notice that [tex]\(-x^2\)[/tex] and [tex]\(+x^2\)[/tex] cancel each other out, so we are left with:
[tex]\[
c^2 = a^2 + b^2 - 2bx
\][/tex]
So, the final simplified equation is:
[tex]\[
c^2 = a^2 + b^2 - 2bx
\][/tex]
This is the equation obtained by considering [tex]\(\triangle CBD\)[/tex] and replacing the expression for [tex]\(h^2\)[/tex].
