Summary
Highlights
A chord is defined as any segment inside a circle that connects two points on the circle's circumference. A diameter is a special type of chord that passes through the center. Arcs are parts of the circle's circumference; a minor arc is less than 180 degrees, while a major arc is greater than 180 degrees and requires three letters to denote.
Two chords are congruent (equal in length) if and only if their corresponding arcs are congruent. This means if arc AB is equal in measure to arc CD, then chord AB is equal in length to chord CD.
Congruent chords are equidistant from the center of the circle. If chord AB is congruent to chord CD, then the distance from the center (E) to AB (EF) is equal to the distance from the center (E) to CD (EG).
If a diameter or a radius is perpendicular to a chord, then it bisects both the chord and its corresponding arc. For example, if EH is perpendicular to AB, then AF equals FB, and arc AH equals arc HB.
Given congruent chords ST and SR, their corresponding arcs must also be congruent. Set the expressions for the arc measures (7x + 24 and 115) equal to each other to solve for x, finding x = 13.
With congruent arc markings, the corresponding chords are equal. Set the chord lengths (9x - 34 and 4x + 1) equal to each other to solve for x. Then, substitute x back into the expression for XY to find its length, XY = 29.
Given corresponding arcs with equal measures and one known arc (76 degrees), utilize the fact that the total degrees in a circle sum to 360. Set up an equation with the sum of all arc measures equal to 360 to find x = 27.
This example combines chord lengths and arc measures. Given that chord RS is 18 and arc TY is 42 degrees, use the theorems to find other lengths and arc measures, like TU being 18 and arc YU being 42 degrees.
In a circle with a radius of 13 and a segment length of 5, use the Pythagorean theorem (a² + b² = c²) to find the missing half of the chord (12). Since the radius is perpendicular to the chord, it bisects it, making the full chord length VW equal to 24.
To find the measure of arc YW, which corresponds to a central angle within a right triangle, use inverse trigonometry. With the adjacent side (5) and the hypotenuse (13) of the right triangle, calculate the inverse cosine to find the central angle, which is 67.4 degrees.