Answer:
Below in bold.
Step-by-step explanation:
If the 3 points are on the circle with the given center then the distance from each point to the center will all be equal. This distance will be the radius of the circle.
Distance of P (2, 5) from the center (4, 2)
= √( (5-2)^2 + (2-4)^2)
= √(9 + 4)
= √13.
For Q (6, -1)this distance is:
= √(6-4)^2 + (-1-2)^2)
= √13.
For R (7, 4) his distance is:
= √(7-4)^2 + (4-2)^2)
= √13
These distances are the same so the points P, Q and R lie on the circle.
The midpoint of PQ
= (2+6)/2 , (5 - 1)/2
= (4, 2).
Which is the center so this verifies the second part.
Answer:
steps below
Step-by-step explanation:
P(2,5) Q(6,-1) R(7,4)
PQ² = (6-2)² + (-1 - 5)² = 16+36 = 52
QR² = (7-6)² + (4 - -1)² = 26
RP² = (7-2)² + (4-5)² = 26
PQ² = QR² + RP²
∴PQR is a right angle triangle and PQ is hypotenuse and it's the diameter of circumscriber triangle
∴P,Q,R are on a circle centered at the mid-point of PQ (4,2) - bisector of PQ
** property of right triangle: For any right triangle, the hypotenuse is a diameter of the circumscribed circle, i.e. the center of the circle is the midpoint of the hypotenuse