Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**Educational Website Content: Geometry Problem**
**Title: Geometry Exercise: Find x**
**Instruction:**
In this problem, you are given a right triangle with an inscribed circle. The sides of the triangle measure 6 units and 4 units, and the small segment along the hypotenuse measures 2 units. Your task is to find the value of \( x \), the length of the hypotenuse.
**Diagram Description:**
The given diagram consists of a right triangle with a circle inscribed within it. Here are the detailed observations:
1. The right triangle has sides labeled as 6 units and 4 units.
2. The inscribed circle touches all three sides of the triangle.
3. There is a segment along the hypotenuse, starting from one point where the circle touches the hypotenuse, labeled as 2 units.
4. The hypotenuse is denoted as \( x \).
The precise mathematical analysis to find the value of \( x \) would involve using the Pythagorean theorem or other geometric properties related to right triangles and inscribed circles.
**Hint:**
Use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
Where \( a \) and \( b \) are the lengths of the legs of the right triangle, and \( c \) is the length of the hypotenuse.
**Solution:**
1. Plug in the values: \( 6^2 + 4^2 = x^2 \).
2. Calculate the squares: \( 36 + 16 = x^2 \).
3. Sum the squares: \( 52 = x^2 \).
4. Find \( x \) by taking the square root of 52:
\[ x = \sqrt{52} = 2\sqrt{13} \]
Thus, the value of \( x \) is \( 2 \sqrt{13} \) units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6756c5c2-608f-4015-b13f-deb910f8c3e1%2F008af64e-7a4d-46b0-8763-db3c93cc7d30%2Fi96i8m_processed.jpeg&w=3840&q=75)
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