Circle Properties - SSAT Middle Level: Quantitative

A chord through the center; length $2r$. A diameter is the longest chord, extending from one point on the circle through the center to the opposite point.

$d = 2r$. The diameter is twice the length of the radius since it spans two radii end to end.

It passes through the center of the circle. The perpendicular bisector of any chord in a circle must pass through the center due to symmetry.

It bisects the chord (cuts it into two equal parts). A radius perpendicular to a chord bisects it because the perpendicular from the center creates two congruent right triangles.

It is perpendicular to the chord. The line from the center to the midpoint of a chord forms the shortest distance and is thus perpendicular to the chord.

They are equidistant from the center. Congruent chords are equidistant from the center because equal lengths imply equal perpendicular distances in the same circle.

The chords are congruent. Chords equidistant from the center are congruent due to the circle's symmetry and the chord length formula.

The diameter is the longest chord. The longest chord is the diameter because it passes through the center, maximizing the distance between two points on the circle.

It is inside the circle (not at the center). The midpoint of a non-diameter chord lies inside the circle but not at the center, as only the diameter's midpoint coincides with the center.

The chord passes through the center. A chord becomes a diameter when it passes through the center, making it the longest possible chord with length $2r$.

The perpendicular distance from the center to the chord. The distance from the center to a chord is the length of the perpendicular segment connecting them.

Chord length $= 2\sqrt{r^2 - d^2}$. The formula derives from the Pythagorean theorem applied to the right triangle formed by the radius, half-chord, and distance $d$.

$14$. The diameter is twice the radius by definition, so $d = 2 \times 7 = 14$.

$9$. The radius is half the diameter by definition, so $r = 18 / 2 = 9$.

A segment from the center to a point on the circle. The radius is the fixed distance from the center to any point on the circumference.

A segment with both endpoints on the circle. A chord connects two points on the circumference of the circle.

Their distances from the center are equal. Equal chord lengths imply equal perpendicular distances from the center in the same circle.

$2\sqrt{69}$. Using the chord length formula, $2\sqrt{13^2 - 10^2} = 2\sqrt{169 - 100} = 2\sqrt{69}$.

$5$. The distance $d = \sqrt{r^2 - (\frac{24}{2})^2} = \sqrt{169 - 144} = \sqrt{25} = 5$.

$6$. The perpendicular radius bisects the chord into two equal halves, so each is $12 / 2 = 6$.

$10$. The chord length is the sum of the two equal segments created by the bisecting perpendicular radius, so $5 + 5 = 10$.

Chord $AB$. Longer chords are closer to the center, so the $16$-unit chord $AB$ is closer than the $10$-unit chord $CD$.

Chord $AB$. Chords closer to the center are longer due to the inverse relationship between distance and chord length.

$16$. Applying the formula $2\sqrt{r^2 - d^2}$, substitute $r=10$ and $d=6$ to get $2\sqrt{100 - 36} = 2\sqrt{64} = 16$.