In the given figure, A, B and C are three points on a circle. If AB = 3 cm and BC = 4 cm, then the measure of the radius of the circle is:
$\frac{5}{2}cm$
To find the radius of the circle given the points \(A\), \(B\), and \(C\) on the circle with \(AB = 3 \text{ cm}\) and \(BC = 4 \text{ cm}\), we must consider the properties of a circle and the triangle formed by these points.
Assuming \(A\), \(B\), and \(C\) are points on the circle, the triangle \(ABC\) forms a right-angled triangle inscribed in the circle, with the hypotenuse as the diameter of the circle. This is based on Thales' theorem.
Since \( \angle ACB = 90^\circ \), we can treat \(AC\) as the hypotenuse of the triangle \(ABC\). Using the Pythagorean theorem:
\(AC^2 = AB^2 + BC^2\)
Substitute the given values:
\(AC^2 = 3^2 + 4^2 = 9 + 16 = 25\)
Therefore, \(AC = \sqrt{25} = 5 \text{ cm}\).
Since \(AC\) is the diameter of the circle (from \(A\) to \(C\)), the radius \(r\) is half of the diameter.
\(r = \frac{AC}{2} = \frac{5}{2} \text{ cm}\)
Thus, the measure of the radius of the circle is \(\frac{5}{2} \text{ cm}\).
The correct answer is \(\frac{5}{2} \text{ cm}\).
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Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
