We'll do it their way, but first let's find OC another way.
AB is the hypotenuse, so AB² = 5² + (5×2)² = 5²(1+2²) = 5³
AB = 5√5
OC is an altitude so the area of the triangle is
(1/2) OA OB = (1/2) OC AB
5(10) = OC (5√5)
OC = 10/√5 = 2√5
OK, we'll check that later.
a)
Equation of AB. Point point form for the join of (a,b) and (c,d) is
(c-a)(y-b)=(d-b)(x-a)
A(0,5), B(10,0)
(10 - 0)(y - 5) = (0 - 5)(x - 0)
10y - 50 = -5x
Answer: AB is x + 2y = 10
OC is perpendicular so we swap the coefficients on x and y, negating one. Through the origin means a zero constant.
Answer: OC is 2x - y = 0
b)
We find the meet, second equation times 2:
4x - 2y = 0
Add first equation,
5x = 10
x = 2
y = 2x = 4
Check 2(2) -4 = 0, good
Answer: C is (2,4)
distance d satisfies
d² = 2² + 4² = 4 + 16 = 20
d = 2√5
Answer: Distance 2√5
That's what we got at the outset. Math works!