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.
what is the value of x?
![This image depicts a right-angled triangle with the following measurements:
- The length of the hypotenuse is labeled as 13 feet.
- One of the legs (the height) is labeled as 5 feet.
- The other leg (the base) is labeled as \( x \).
### Explanation of the Triangle:
- **Right Angle:** The triangle has a right angle (90 degrees) at the bottom right corner, indicated by the small red square.
- **Hypotenuse:** The side opposite the right angle, labeled as 13 feet, is the longest side of the triangle.
- **Height:** The leg that is perpendicular to the base is labeled as 5 feet.
- **Base:** The base of the triangle is labeled as \( x \), which is what we need to solve for.
### Solving for \( x \):
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
The formula is:
\[ c^2 = a^2 + b^2 \]
In this case:
\[ 13^2 = x^2 + 5^2 \]
Solving for \( x \):
\[ 169 = x^2 + 25 \]
\[ x^2 = 169 - 25 \]
\[ x^2 = 144 \]
\[ x = \sqrt{144} \]
\[ x = 12 \]
So, the length of the base \( x \) is 12 feet.
### Educational Value:
This example is perfect for illustrating the application of the Pythagorean theorem. It teaches students how to identify the components of a right-angled triangle and use the theorem to find a missing side length.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56711c57-cc3c-4bb4-ba0b-c8c4ac634c9a%2Ff2a30e0c-c71c-4552-8477-0123ba0fcf20%2F0ngy6rd_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 1 images
