Circle Properties - SSAT Middle Level Quantitative

1. What is the property of the segment from the center to the midpoint of a chord?

   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.

2. In the same circle, chord $AB$ is closer to the center than chord $CD$. Which chord is longer?

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

3. In the same circle, chord $AB$ is $16$ and chord $CD$ is $10$. Which chord is closer to the center?

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

4. What is the property of a radius drawn perpendicular to a chord?

   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.

5. Find the radius if the diameter is $18$ units.

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

6. Find the diameter if the radius is $7$ units.

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

7. What is the chord-length formula in terms of radius $r$ and center distance $d$?

   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 $d$.

8. What is the definition of a radius of a circle?

   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.

9. What is the distance from the center to a chord called (in circle-chord problems)?

   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.

10. A circle has radius $10$. A chord is $6$ units from the center. Find the chord length.

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

11. What is the relationship between a radius and a diameter in any circle?

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

12. A radius is perpendicular to a chord and bisects it into segments of $5$ and $5$. What is the chord length?

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

13. What is true about the longest chord in a circle?

    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.

14. Which chord is longer: one closer to the center or one farther from the center?

    The chord closer to the center is longer. A chord closer to the center is longer because the perpendicular distance $d$ is smaller, increasing $2\sqrt{r^2 - d^2}$.

15. Identify the condition that makes a chord a diameter.

    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 $2r$.

16. A radius is perpendicular to a chord of length $12$. What is each half of the chord?

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

17. What is the definition of a chord in a circle?

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

18. What is true about the midpoint of any chord that is not a diameter?

    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.

19. What is the definition of a diameter of a circle?

    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.

20. What is true if two chords are equidistant from the center of 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.

21. What is the perpendicular bisector property for a chord in a circle?

    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.

22. Two chords in the same circle are each $12$ units long. What can you conclude about their distances from the center?

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

23. A chord is $24$ units long in a circle of radius $13$. Find the distance from the center to the chord.

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

24. A circle has radius $13$. A chord is $10$ units from the center. Find the chord length.

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

25. What is true about congruent chords in the same circle?

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

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