Question

the end points of the diameter of a circle are at the points a (3, 5) and b (5, 6). what is the length of the radius? give your answer correct to 2 decimal places if necessary.
a) 2.5 units
b) 5 units
c) 3.77 units
d) 1.12 units

Explanation:

Step1: Find the distance between A and B (diameter)

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For points A\((3, 5)\) and B\((5, 6)\), we have \(x_1 = 3\), \(y_1 = 5\), \(x_2 = 5\), \(y_2 = 6\). Plugging into the formula:
\(d=\sqrt{(5 - 3)^2 + (6 - 5)^2}=\sqrt{2^2 + 1^2}=\sqrt{4 + 1}=\sqrt{5}\approx2.236\)? Wait, no, wait: Wait, \((5 - 3)=2\), \((6 - 5)=1\), so \((2)^2=4\), \((1)^2=1\), sum is \(5\), square root of \(5\) is approximately \(2.236\)? Wait, no, wait the options have 1.12? Wait, maybe I miscalculated. Wait, no, radius is half of diameter. Wait, let's recalculate the distance between A(3,5) and B(5,6).

\(x_2 - x_1 = 5 - 3 = 2\)

\(y_2 - y_1 = 6 - 5 = 1\)

So distance \(d = \sqrt{(2)^2 + (1)^2}=\sqrt{4 + 1}=\sqrt{5}\approx2.236\)? But that's the diameter? Wait, no, wait the options: a) 2.5, b)5, c)3.77, d)1.12. Wait, maybe I made a mistake. Wait, no, maybe the points are A(3,5) and B(5,6)? Wait, no, maybe the coordinates are different? Wait, no, let's check again. Wait, maybe the points are A(3,5) and B(5, 6)? Wait, no, maybe I mixed up x and y. Wait, no, the distance formula is correct. Wait, but \(\sqrt{5}\approx2.236\), so radius would be half of that, which is \(\frac{\sqrt{5}}{2}\approx1.118\approx1.12\), which is option d. Wait, but let's recalculate the distance between A and B. Wait, maybe the points are A(3,5) and B(5, 6)? Wait, 5-3=2, 6-5=1, so distance is \(\sqrt{2^2 + 1^2}=\sqrt{5}\approx2.236\), so radius is half, so \(\frac{\sqrt{5}}{2}\approx1.12\), which is option d. Wait, but the initial thought was wrong. Let's do it step by step.

Step1: Calculate the distance between points A(3,5) and B(5,6) (diameter length)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Substitute \(x_1 = 3\), \(y_1 = 5\), \(x_2 = 5\), \(y_2 = 6\):

\(d = \sqrt{(5 - 3)^2 + (6 - 5)^2}\)

\(d = \sqrt{(2)^2 + (1)^2}\)

\(d = \sqrt{4 + 1}\)

\(d = \sqrt{5} \approx 2.236\)

Step2: Calculate the radius (half of diameter)

Radius \(r = \frac{d}{2}\)

\(r = \frac{\sqrt{5}}{2} \approx \frac{2.236}{2} \approx 1.118 \approx 1.12\) (rounded to two decimal places)

Answer:

d) 1.12 units