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.
![### Find the Values of x and y in the Following Right Triangle
In the given problem, you are presented with a right triangle. The following properties are provided:
1. There is a right angle, indicating that one of the angles is 90 degrees.
2. One of the other angles is given as 45 degrees.
3. The hypotenuse of the triangle is labeled as 20 units.
The sides `x` and `y` of the triangle are unknown and you are required to find their values. The triangle diagram is as follows:
```
/|
/ |
/ |
y / | 20
/__ |
45° |____|
x
```
Below the diagram, there are two input boxes where you need to provide the values of `x` and `y`:
```
x = [ ]
y = [ ]
```
### Explanation
Since one angle is 45 degrees and the other non-right angle is also 45 degrees (because the sum of angles in a triangle is 180 degrees), you have an isosceles right triangle (also known as a 45-45-90 triangle).
In a 45-45-90 triangle, the lengths of the legs (the sides opposite the 45-degree angles) are equal, and the hypotenuse is equal to the length of one leg multiplied by √2 (square root of 2).
Given that the hypotenuse is 20 units, you can use the formula for a 45-45-90 triangle:
\[ \text{leg} \times \sqrt{2} = \text{hypotenuse} \]
\[ \text{leg} \times \sqrt{2} = 20 \]
\[ \text{leg} = \frac{20}{\sqrt{2}} \]
\[ \text{leg} = \frac{20 \sqrt{2}}{2} = 10 \sqrt{2} \]
Therefore:
\[ x = 10 \sqrt{2} \]
\[ y = 10 \sqrt{2} \]
**In the input boxes, you should enter:**
```
x = 10√2
y = 10√2
```](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7c29d2f-5cf3-43f5-aa90-abc5a822140a%2F63e74266-d701-4908-928e-6e1d59bece39%2Fxkmsrjv_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 1 images
