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.
In the given right triangle, we have:
- The length of one leg as \( a \).
- The length of the other leg as \( b \).
- The hypotenuse has a length of \( 4\sqrt{2} \).
- One of the angles is \( 45^\circ \).
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**Part A**
Find the value of \( a \)
Options:
1. \( \frac{4}{\sqrt{2}} \)
2. \( 4\sqrt{2} \)
3. \( 4 \)
4. \( \frac{4}{2} \)
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**Explanation:**
The given diagram represents a right triangle with a \( 45^\circ \) angle, making it a 45-45-90 triangle. In such triangles, the legs are congruent, and the hypotenuse is \( \sqrt{2} \) times the length of each leg.
Given the hypotenuse is \( 4\sqrt{2} \), we can set up the following relationship:
\[ a\sqrt{2} = 4\sqrt{2} \]
Solving for \( a \):
\[ a = 4 \]
Therefore, the value of \( a \) is \( 4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34bff1e1-af68-4229-b0ef-aa0aaa134e46%2F8300e3b3-5df9-4051-a1e0-57b5fa5c45a2%2Fttbavnv_processed.jpeg&w=3840&q=75)
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