#1**+2 **

First, we realize that the diameter of our given circle O is 10.

That means that the length between O and the intersection with the chord is 3, because:

10-(5+2) = 10 -7 = 3

If we draw two radii to the endpoints of the chord, we get two congruent right triangles by HL congruency(their hypotenuses are both radii of the circle, and they share a common leg). The hypotenuse of one of these right triangles is 5, and their leg is 3(which we just got). To solve for the value of x, we can then write by the pythagorean theorem:

\(5^2 = 3^2 + x^2\)

\(25 = 3^2 + x^2\)

\(25 = 9 + x^2\)

\(16 = x^2\)

Square rooting both sides and discarding the negative solution(a length can't be negative), we arrive at our answer of:

\(x = 4\)

Alternatively, we can realize that the triangle created is a 3-4-5 triangle(through memorization of right triangles), which would be a faster way to calculate the length without going into pythagorean.

jfan17 Apr 18, 2020

#2**+2 **

You could use intersecting chord theorem:

the diameter chord has 2 segments 8 and 2

the unknown chord has two 2 segments x and x

8*2 = x*x

16 = x^2 x = 4

ElectricPavlov Apr 18, 2020