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.
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### Educational Content on Triangle Properties
**Problem Statement:**
**A)** The coordinates of the vertices of triangle \( \Delta ABC \) are \( A(1,2) \), \( B(-5,3) \), and \( C(-6,-3) \). Prove that \( \Delta ABC \) is isosceles.
**B)** Prove \( \Delta ABC \) is a right triangle.
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**Detailed Steps and Diagram Analysis:**
**Graph Description:**
- The image includes a coordinate plane with the \( x \)-axis and \( y \)-axis clearly marked. The grid is used to plot the points \( A(1,2) \), \( B(-5,3) \), and \( C(-6,-3) \).
**Step-by-Step Solution:**
1. **Finding the Length of Sides:**
To prove \( \Delta ABC \) is isosceles, we need to show that at least two sides of the triangle are of equal length.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
* Calculate \( AB \):
\[
AB = \sqrt{((-5) - 1)^2 + (3 - 2)^2} = \sqrt{(-6)^2 + (1)^2} = \sqrt{36 + 1} = \sqrt{37}
\]
* Calculate \( BC \):
\[
BC = \sqrt{((-6) - (-5))^2 + ((-3) - 3)^2} = \sqrt{(-1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37}
\]
* Calculate \( AC \):
\[
AC = \sqrt{((-6) - 1)^2 + ((-3) - 2)^2} = \sqrt{(-7)^2 + (-5)^2} = \sqrt{49 + 25} = \sqrt{74}
\]
Since \( AB = BC = \sqrt{37} \), \( \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc23ada6-c82e-455f-b66b-5f630d4df728%2F2da1442b-0f74-428a-abfb-d01be39b5073%2Fb35p40u_processed.jpeg&w=3840&q=75)
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