Question

1. o is the centre of the circle. determine the value of f to the nearest tenth, if necessary. (circle diagram with o, t, s, u, and lengths 20, 6, f) a. 4 b. 8 c. 64 d. 11.7
2. o is the centre of the circle. determine the value of s to the nearest tenth, if necessary. (circle diagram with o, k, h, j, and lengths 7, s, 10) a. 3 b. 7.1 c. 12.2 d. 51
3. o is the centre of this circle. determine the value of a°. (circle diagram with o, x, y, z, and angle 86°) a. 47° b. 86° c. 94° d. 90°

questions i should practice: /3

Explanation:

Question 26

Step1: Recall the circle theorem (perpendicular from center to chord bisects the chord and use Pythagoras)

The radius of the circle is 20 (since the line from O to the circumference is radius). The distance from O to the chord (OT) is 6. Let half of the chord length be \( x \), then by Pythagoras: \( x = \sqrt{20^2 - 6^2} \). But wait, actually, the length \( f \) here? Wait, maybe the chord is SU, and OT is perpendicular to SU, so OT = 6, radius OS = 20. Wait, no, maybe the segment from O to T is 6, and the radius is 20? Wait, no, maybe the length from O to the chord is 6, and the radius is 20? Wait, no, maybe the given length is the radius? Wait, the diagram has a line from O to S (radius) as 20? Wait, no, maybe the horizontal segment from O to T is 6, and the vertical segment from T to S is \( f \), and OS is the radius. Wait, actually, in a circle, if a perpendicular is drawn from the center to a chord, then it bisects the chord. Also, the radius, the distance from center to chord, and half the chord form a right triangle. Wait, maybe the length from O to T is 6, and the radius is 20? Wait, no, maybe the given 20 is the radius, and 6 is the distance from O to the chord. Then, the length of the chord segment (from T to S) can be found by Pythagoras: \( f = \sqrt{20^2 - 6^2} \)? Wait, no, that would be if OT is 6, OS is 20, then TS is \( \sqrt{20^2 - 6^2} = \sqrt{400 - 36} = \sqrt{364} \approx 19.1 \), but that's not one of the options. Wait, maybe I got it wrong. Wait, the options are 4, 8, 64, 11.7. Wait, maybe the radius is 10? No, the diagram has 20. Wait, maybe the length from O to the chord is 6, and the radius is, say, 10? No, the given is 20. Wait, maybe the segment from O to T is 6, and the length of OT is 6, and the radius is 10? No, the options include 11.7. Wait, let's recalculate: \( \sqrt{20^2 - 6^2} = \sqrt{400 - 36} = \sqrt{364} \approx 19.1 \), not matching. Wait, maybe the given 20 is the diameter? So radius is 10. Then \( \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \). Ah! That makes sense. So if the diameter is 20, radius is 10. Then distance from O to T is 6, so half the chord length is \( \sqrt{10^2 - 6^2} = 8 \). So \( f = 8 \)? Wait, the options have b. 8. So that's the answer.

Step1: Assume the given 20 is the diameter, so radius \( r = 10 \).

Step2: Distance from center O to chord (OT) is 6. Let half the chord length be \( f \) (wait, maybe TS is \( f \), and OT is 6, radius 10). Then by Pythagoras: \( f = \sqrt{10^2 - 6^2} \)

\( f = \sqrt{100 - 36} = \sqrt{64} = 8 \)

Step1: Recall the circle theorem (perpendicular from center to chord bisects the chord)

The diagram shows a chord HJ, with OK perpendicular to HJ, so HK = KJ = 7 (since K is the midpoint). The distance from O to K is \( s \), and the length from O to the chord's side? Wait, the length from O to the chord (OK) is \( s \), and the segment from O to the other side (maybe the radius? Wait, the vertical segment from O to the chord is 10? No, the diagram has 10 as the length from O to the chord? Wait, no, maybe the radius is, let's see. The chord HJ has HK = 7, so HJ = 14. The distance from O to the chord is \( s \), and the length from O to the chord's line is 10? Wait, no, maybe the radius is \( \sqrt{s^2 + 7^2} \), and the other segment is 10? Wait, no, the options include 7.1. Wait, let's think again. Maybe the length from O to the chord is \( s \), and the radius is 10? No, wait, the diagram has a line from O to the chord (OK) as \( s \), and the segment from K to J is 7, and the length from O to the other side (maybe the diameter? No, the options are 3, 7.1, 12.2, 51. Wait, let's use Pythagoras. Suppose the radius is \( R \), the distance from center to chord is \( s \), and half the chord length is 7 (since HK = 7, so HJ is bisected by OK, so HK = KJ = 7). Then, if the length from O to the chord is \( s \), and the radius is, say, the hypotenuse. Wait, maybe the length from O to the chord is \( s \), and the segment from O to the circumference is 10? No, the options have 7.1. Wait, let's calculate: if the radius is, say, the hypotenuse, and one leg is 7 (half the chord), and the other leg is \( s \), and the other segment is 10? Wait, no, maybe the length from O to the chord is \( s \), and the length from K to the other side (along the radius) is 10? Wait, no, maybe the radius is \( \sqrt{s^2 + 7^2} \), and the length from O to the chord's line is 10? No, this is confusing. Wait, the options include 7.1. Let's try \( s = \sqrt{10^2 - 7^2} \)? No, that would be \( \sqrt{100 - 49} = \sqrt{51} \approx 7.1 \). Ah! So if the length from O to the chord (OK) is \( s \), and the segment from O to the circumference (radius) is 10, and half the chord length is 7, then by Pythagoras: \( s = \sqrt{10^2 - 7^2} = \sqrt{51} \approx 7.1 \). So that's option b.

Step1: Let the radius be 10, half the chord length (HK) be 7.

Step2: By Pythagoras, \( s = \sqrt{10^2 - 7^2} \)

\( s = \sqrt{100 - 49} = \sqrt{51} \approx 7.1 \)

Step1: Recall the circle theorem (central angle and inscribed angle)

The central angle \( \angle YOZ = 86^\circ \). The inscribed angle subtended by the same arc YZ would be half the central angle, but wait, \( \angle XYZ \) is an inscribed angle? Wait, no, \( \angle XZY \) or \( \angle XYZ \)? Wait, the diagram has triangle OYZ with \( \angle YOZ = 86^\circ \), and angle at X is \( a^\circ \). Wait, the angle at X ( \( \angle X \)) and the angle at O ( \( \angle YOZ \)): since \( \angle YOZ \) is a central angle, and \( \angle X \) is an inscribed angle subtended by arc YZ. But wait, the sum of angles in a triangle? No, in a circle, the central angle is twice the inscribed angle subtended by the same arc. But here, \( \angle YOZ = 86^\circ \), and \( \angle X \) and \( \angle YOZ \): wait, maybe the angle at X is \( a^\circ \), and the angle at O is \( 86^\circ \), and since O is the center, OX and OZ are radii, OY and OZ are radii. Wait, maybe \( \angle XZY = a^\circ \), and \( \angle YOZ = 86^\circ \). Then, the angle at X: wait, the sum of angles in a triangle? No, the central angle \( \angle YOZ = 86^\circ \), so the inscribed angle subtended by arc YZ would be \( \frac{180^\circ - 86^\circ}{2} \)? Wait, no, maybe the angle at X is supplementary to half of \( \angle YOZ \)? Wait, no, let's think again. The central angle \( \angle YOZ = 86^\circ \), so the arc YZ is \( 86^\circ \). The angle at X ( \( \angle X \)) is an inscribed angle subtended by arc YZ? No, maybe the angle at X and the angle at O: since \( OX = OZ \) (radii), triangle OXZ is isoceles. Wait, no, maybe the angle at X is \( a^\circ \), and the angle \( \angle YOZ = 86^\circ \), so the reflex angle at O is \( 360^\circ - 86^\circ = 274^\circ \), no. Wait, the sum of angles in a triangle: if O is the center, then \( OX = OZ = OY \) (radii). So triangle OYZ is isoceles with OY = OZ. Wait, no, the angle at O is \( 86^\circ \), so the base angles at Y and Z are \( \frac{180^\circ - 86^\circ}{2} = 47^\circ \). Then, angle at X: maybe \( a^\circ = 47^\circ \)? Wait, the options have a. \( 47^\circ \), b. \( 86^\circ \), c. \( 94^\circ \), d. \( 90^\circ \). Wait, maybe the angle at X is \( 47^\circ \), because the central angle is \( 86^\circ \), so the inscribed angle is half of the supplementary angle? Wait, no, the central angle \( \angle YOZ = 86^\circ \), so the inscribed angle subtended by arc YZ is \( \frac{1}{2} \times (180^\circ - 86^\circ) \)? No, that's not right. Wait, the correct approach: the central angle \( \angle YOZ = 86^\circ \), so the arc YZ is \( 86^\circ \). The angle at X ( \( \angle X \)) is an inscribed angle subtended by arc YZ, but no, arc YZ is \( 86^\circ \), so the inscribed angle would be \( 43^\circ \), but that's not an option. Wait, maybe the angle at X and the angle at O: since \( \angle YOZ = 86^\circ \), then the angle \( \angle X \) is \( \frac{180^\circ - 86^\circ}{2} = 47^\circ \)? Wait, yes! Because in triangle OYZ, \( OY = OZ \) (radii), so it's isoceles, so angles at Y and Z are \( (180 - 86)/2 = 47^\circ \). Then, angle at X: maybe \( \angle X = 47^\circ \), which is option a.

Step1: Central angle \( \angle YOZ = 86^\circ \), triangle OYZ is isoceles (OY = OZ).

Step2: Calculate base angles: \( \angle OYZ = \angle OZY = \frac{180^\circ - 86^\circ}{2} = 47^\circ \). But angle at X ( \( a^\circ \)) is equal to \( \angle OZY \)? Wait, maybe because of alternate segments or something. Wait, the answer is \( 47^\circ \), option a.

Answer:

b. 8

Question 27