Find the Radius of a Quarter-Circle Using Right Triangles

Math Problem Statement

Find the radius of the quarter-circle.

[asy] unitsize(4 cm); pair A, B, C, D, O; A = (1,0); B = dir(60); D = (-1,0); C = extension(B, D, O, (0,1)); draw((0,1)--O--A); draw(A--B--C--cycle); draw(rightanglemark(A,B,C,2)); draw(arc(O,1,0,90)); draw(rightanglemark((0,1),(0,0),A,2)); label("$A$", A, SE); label("$B$", B, B); label("$C$", C, W); label("$O$", O, SW); label("$3$", (B + C)/2, NW, red); label("$4$", (A + B)/2, NE, red); [/asy] Hint(s): When we see right angles in circles, what should we think of?

Solution

We are given a right-angled triangle inscribed in a quarter-circle, with points $A$ , $B$ , $C$ , and $O$ , where $O$ is the center of the circle. The lengths of $AB = 4$ and $BC = 3$ are provided, and we need to find the radius of the circle.

Since the triangle $ABC$ is a right triangle, and the right angle is at $B$ , the hypotenuse $AC$ is the diameter of the circle. The Pythagorean theorem can be applied to the triangle $ABC$ :

$AC^2 = AB^2 + BC^2$

Substituting the given values:

$AC^2 = 4^2 + 3^2 = 16 + 9 = 25$

Thus:

$AC = \sqrt{25} = 5$

Since $AC$ is the diameter of the quarter-circle, the radius $r$ is half of the diameter:

$r = \frac{AC}{2} = \frac{5}{2} = 2.5$

So, the radius of the quarter-circle is $2.5$ .

Would you like more details on this solution, or do you have any questions?

How would you calculate the length of the hypotenuse in another right-angled triangle?
What is the relationship between the radius and the diameter of a circle?
Can you apply the Pythagorean theorem to other geometric shapes?
How does this scenario relate to circles in coordinate geometry?
What is the significance of inscribed angles in a circle?

Tip: When dealing with right-angled triangles inscribed in circles, remember that the hypotenuse is always the diameter.

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Math Problem Analysis

Mathematical Concepts

Geometry
Right Triangles
Circles
Pythagorean Theorem

Formulas

Pythagorean theorem: a^2 + b^2 = c^2
Relationship between radius and diameter: radius = diameter / 2

Theorems

Pythagorean Theorem
Theorem of Right Triangles Inscribed in a Circle

Suitable Grade Level

Grades 8-10

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