question

What must be the length of ZY in order for ZY to be tangent to circle X at point Y?

answer

15 units

**Explanation:** The length of ZY must be equal to the radius of circle X.

question

Angles PTQ and STR are vertical angles and congruent.
Which chords are congruent?

answer

QP and SR

**Explanation:** The chords of a circle are congruent if and only if they subtend the same angle at the center of the circle. In this case, the angles PTQ and STR are vertical angles, which means they subtend the same angle at the center of the circle. Therefore, the chords PT and ST are congruent.

question

In circle N, KL ≅ ML. Circle N is shown. Line segments N J, N M, N L, and N K are radii. Lines are drawn to connect each point on the circle to create secants J M, M L, L K, and K J. M L and K L are congruent. The measure of arc J K is (5 x + 24) degrees, the measure of arc J M is (13 x + 2) degrees, the measure of arc M L is (8 x minus 3) degrees, and the measure of arc K L is (7 x + 7) degrees. What is the measure of Arc J M?

answer

132°

**Explanation:** The measure of arc J M is (13 x + 2) degrees.

question

In circle D, ∠EDH ≅ ∠EDG. Circle D is shown. Line segment F H is a diameter. Line segments D E and D G are radii. Lines are drawn from point E to point G and from point E to point H to form secants. The length of E J is 4 and the length of E H is 9. The measure of arc F E is 57 degrees and the measure of arc F G is 66 degrees. What is the measure of Arc E H?

answer

123°

**Explanation:** In circle D, ?EDH ? ?EDG. This means that the measure of arc EH is equal to the measure of arc EG. We can find the measure of arc EH by subtracting the measure of arc EG from the measure of the whole circle, which is 360 degrees. The measure of arc EG is 66 degrees, so the measure of arc EH must be 360 - 66, or 294 degrees.

question

QR is tangent to circle P at point Q.
What is the approximate length of RP? Round to the nearest tenth.

answer

6.1 units

**Explanation:** The length of RP is approximately 10.7.

question

Circle D is shown with the measures of the minor arcs.
Which angles are congruent?

answer

∠GDH and ∠EDH

**Explanation:** The angles that are congruent are the ones that have the same measure. In this case, that would be angles A and C, since they are both 60 degrees.

question

In circle A, ∠BAE ≅ ∠DAE.
What is the value of x?

answer

17

**Explanation:** Given that angle BAE is equal to angle DAE, we can say that angles BAC and DAF are also equal. This is because angles BAE, BAC, and DAE form a linear pair, meaning that the sum of the angles is 180 degrees. The same is true for angles DAE, DAF, and DAC. This means that angle BAC is equal to angle DAF, and angle DAC is equal to angle BAE. Therefore, angle BAC is equal to angle DAF, and angle DAC is equal to angle BAE. We can also say that angle BAC is equal to angle DAE, since they are both part of a linear pair. Thus, we can conclude that angle BAC is equal to angle DAE, and angle DAC is equal to angle BAE. Therefore, the value of x is equal to angle BAC, which is equal to angle DAE.

question

is tangent to circle G at point F. Circle G is shown. Line segment G F is a radius. Line segment F H is a tangent that intersects the circle at point F. A line is drawn from point H to point G and goes through a point on the circle. The length of F H is 35, and the length of the line segment from point H to the point on the circle is 25. The lengths of the radii are r. What is the length of the radius, r?

answer

12 units

**Explanation:** The length of the radius, r, is 35.

question

Angles PTQ and STR are vertical angles and congruent.
Which arcs are congruent?

answer

NOT A

**Explanation:** The arcs PT and ST are congruent because they are both part of the circumference of the circle and they are both subtended by the same central angle.

question

What is true regarding two adjacent arcs created by two intersecting diameters?

answer

The sum of their measures is 180°.

**Explanation:** The two arcs created by two intersecting diameters will always be equal in length. This is because the diameter is the longest chord of a circle, and the two arcs are created by two diameters that intersect at the center of the circle.