Question

I need help with these questions (see image). Please show workings.​

Answer 1

r =
`(1)/(2)`
`√(l^2+4h^2)`

Explanation:

h bisects l at right angles.

There is a right triangle formed with legs h and
`(1)/(2)` l and hypotenuse r

Using Pythagoras' identity, then

r² = (
`(1)/(2)` l )² + h²

=
`(1)/(4)` l² + h² ← factor out
`(1)/(4)`

=
`(1)/(4)` (l² + 4h²)

Take the square root of both sides

r =
`\sqrt{(1)/(4)(l^2+4h^2) }` =
`(1)/(2)`
`√(l^2+4h^2)`

Answer 2

• r =
  `√((l/2)^2 + h^2)`

Explanation:

Connect the endpoints of the with the center. The formed triangle is isosceles with sides l, r and r.

The height of the triangle h, is the distance between the center of the circle and the midpoint of the chord.

As per Pythagorean theorem we have:

• (l/2)² + h² = r²
• 
  `√((l/2)^2 + h^2)` = r
• r =
  `√((l/2)^2 + h^2)`