5.2 Inner product (applications)

Consider a triangle [math]ABC[/math] with vertices [math]A(3,5), B(-3,7)[/math] and [math]C(-1,1)[/math]. Find the coordinates of the center of [math]ABC[/math]’s circumcircle.

If the distance between point [math]P(1,3)[/math] and line [math]L: x+y=6[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is a rational number, what is the value of [math]a[/math]?

Find the distance between two parallel lines [math]3x-4y=7[/math] and [math]3x-4y=3[/math].

Let [math]a, b[/math], and [math]c[/math] be three real numbers satisfying [math]a+b+c=7[/math]. Find [math]a, b[/math], and [math]c[/math] to minimize [math]a^2+4b^2+9c^2[/math].

Consider a triangle [math]ABC[/math] that satisfies [math]a=5, b=7, c=8[/math]. Let [math]P[/math] be a point inside [math]ABC[/math]. Suppose for triangles [math]BPC, APC[/math], and [math]APB[/math], [math]h_a, h_b[/math], and [math]h_c[/math] are the heights on sides [math]\overline{BC}, \overline{AC}[/math], and [math]\overline{AB}[/math] respectively. If the minimum of [math]\dfrac{5}{h_a}+\dfrac{7}{h_b}+\dfrac{8}{h_c}[/math] is [math]a\sqrt{3}[/math] where [math]a[/math] is a rational number, what is the value of [math]a[/math]?

Consider line [math]L: 2x+y-5=0[/math] and two points [math]P(1,-7)[/math] and [math]Q(-2,3)[/math]. Let [math]R[/math] be a point on [math]L[/math]. If the minimum of [math]PR+RQ[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

Let [math]A(3,5), B(-4,4)[/math], and [math]C(0,1)[/math] be three points. Find the equation of the circumcircle of triangle [math]ABC[/math]. (The equation of a circle with radius [math]r[/math] and center [math](x_0, y_0)[/math] is [math](x-x_0)^2+(y-y_0)^2=r^2[/math]).