Question

15.
o is the center of the circle. assume that lines that appear to be tangent are tangent. what is the value of x?
(the image shows a circle with center o, a 124° central angle, and two tangent lines from a common external point forming an angle of x° with each other.)

Explanation:

Step1: Find the central angle supplementary to 124°

The sum of angles around a point (center of the circle) is 360°, but here we have a straight line (diameter - related) and the given 124° angle. Wait, actually, the tangent to a circle is perpendicular to the radius at the point of contact. So the two radii (from center O to the points of tangency) are perpendicular to the tangent lines. So the quadrilateral formed by the two tangent lines and the two radii has two right angles (90° each) at the points of tangency, one angle at the center (let's call it \( \theta \)) and the angle \( x \) at the external point. The sum of interior angles of a quadrilateral is 360°. First, find the central angle \( \theta \) that is supplementary to 124°? Wait, no. Wait, the given angle at the center is 124°, but actually, the two radii and the line through the center form a straight line? Wait, no, the angle between the two radii: wait, the total around the center is 360°, but the angle between the two radii (the ones to the points of tangency) and the given 124° angle: wait, the straight line through the center would be 180°, but here the given angle is 124°, so the angle between the two radii (let's call it \( \alpha \)) is \( 180° - 124° \)? No, wait, no. Wait, the two tangent lines from an external point to a circle are equal in length, and the line from the external point to the center bisects the angle between the tangents and the angle between the two radii. Also, the radius is perpendicular to the tangent, so each radius - tangent angle is 90°. So the quadrilateral has two right angles (90° each), one angle at the center (let's say \( \beta \)) and the angle \( x \) at the external point. So sum of angles in quadrilateral: \( 90° + 90° + \beta + x = 360° \), so \( \beta + x = 180° \). Now, what is \( \beta \)? The angle at the center between the two radii. The given angle is 124°, and since the total around the center is 360°, but actually, the angle \( \beta \) and 124° are supplementary? Wait, no, the straight line through the center: the angle on a straight line is 180°, so if one part is 124°, the other part (the angle between the two radii, \( \beta \)) is \( 180° - 124° = 56° \)? Wait, no, that's not right. Wait, the center O, the two radii are to the points of tangency, and the line from O to the external point. Wait, maybe I made a mistake. Let's start over.

The formula for the angle between two tangents from an external point: the measure of the angle between two tangents drawn from an external point to a circle is equal to the difference between 180° and the measure of the central angle between the two radii to the points of tangency. Wait, no, the correct formula is: if \( x \) is the angle between the tangents, and \( \theta \) is the central angle between the radii, then \( x + \theta = 180° \)? No, wait, the sum of the angle between the tangents (x) and the central angle (θ) is 180°? Wait, no, the two radii are perpendicular to the tangents, so each radius - tangent angle is 90°, so the quadrilateral has two 90° angles, one angle x (between tangents) and one angle θ (central angle between radii). So sum of angles in quadrilateral: 90 + 90 + θ + x = 360 ⇒ θ + x = 180. Now, what is θ? The central angle between the two radii. The given angle is 124°, and since the total around the center is 360°, but the angle θ and 124° are supplementary? Wait, no, the angle 124° and θ are on a straight line? Wait, the center O, the line through O: the angle on a straight line is 180°, so if one angle is 124°, the other angle (θ) is 180° - 124° = 56°? Wait, no, that's not. Wait, the central angle between the two radii is θ, and the given angle is 124°, so θ = 180° - 124°? No, that's incorrect. Wait, the total around the center is 360°, but the two radii and the line through O: no, the two radii are to the points of tangency, and the line from O to the external point. Wait, maybe the given angle of 124° is the reflex angle? No, the diagram shows a 124° angle at O, between a radius and a line. Wait, let's look at the diagram: O is the center, there is a line from O to a point on the circle, making 124° with another line? Wait, the two tangent lines from the external point to the circle, so the two radii (from O to each tangent point) are perpendicular to the tangents. So the quadrilateral has two right angles (90° each), one angle at O (let's call it \( \alpha \)) and the angle \( x \) at the external point. So sum of angles: 90 + 90 + \( \alpha \) + x = 360 ⇒ \( \alpha \) + x = 180. Now, \( \alpha \) is the central angle between the two radii. The given angle is 124°, and since \( \alpha \) and 124° are supplementary (because they form a linear pair, i.e., a straight line), so \( \alpha = 180° - 124° = 56° \)? Wait, no, that would mean \( \alpha = 56° \), then x = 180° - 56° = 124°? No, that can't be. Wait, I think I messed up the central angle. Wait, the correct formula: the measure of the angle between two tangents drawn from an external point to a circle is equal to 180° minus the measure of the central angle between the two radii to the points of tangency. So if \( x \) is the angle between the tangents, and \( \theta \) is the central angle, then \( x = 180° - \theta \). Now, what is \( \theta \)? The central angle between the two radii. The given angle is 124°, and since the total around the center is 360°, but the angle \( \theta \) and 124° are supplementary? Wait, no, the angle 124° is adjacent to \( \theta \) and they form a linear pair (180°), so \( \theta = 180° - 124° = 56° \)? No, that's not. Wait, the central angle \( \theta \) is actually equal to 180° - 124°? Wait, no, let's calculate again.

Wait, the two tangent lines: each is perpendicular to the radius, so the radius - tangent angle is 90°. So the quadrilateral has two 90° angles, one angle at O (let's say \( \theta \)) and one angle at the external point (x). So sum of angles in quadrilateral: 90 + 90 + \( \theta \) + x = 360 ⇒ \( \theta + x = 180 \). Now, \( \theta \) is the central angle between the two radii. The given angle is 124°, and since \( \theta \) and 124° are supplementary (because they are on a straight line), so \( \theta = 180° - 124° = 56° \). Then, substituting into \( \theta + x = 180 \), we get \( 56° + x = 180° \), so \( x = 180° - 56° = 124° \)? No, that's not right. Wait, I think I have the central angle wrong. Wait, the central angle between the two radii is actually equal to 180° - 124°? No, that's incorrect. Wait, the total around the center is 360°, but the angle 124° and the central angle \( \theta \) are related as \( \theta = 360° - 124° \)? No, that would be 236°, which is too big. Wait, no, the diagram shows a 124° angle at O, between a radius and a line. Wait, maybe the 124° angle is the angle between the line through O and one of the radii, and the other radius is on the other side, forming a straight line? Wait, no, the two radii are to the points of tangency, so the angle between them is \( \theta \), and the given angle is 124°, so \( \theta = 180° - 124° = 56° \). Then, the angle between the tangents: since the two radii are perpendicular to the tangents, the quadrilateral has two right angles, so the angle between the tangents (x) and the central angle (θ) add up to 180°, so x = 180° - θ = 180° - 56° = 124°? No, that can't be. Wait, I think I made a mistake in the quadrilateral. Wait, the two tangent lines from an external point: the angle between them (x) and the central angle (θ) are related by \( x = 180° - θ \), where θ is the central angle between the two radii. Now, what is θ? The given angle is 124°, and since θ and 124° are supplementary (they form a linear pair), θ = 180° - 124° = 56°. Then x = 180° - 56° = 124°? No, that's not correct. Wait, no, the formula is that the angle between two tangents is equal to 180° minus the central angle. Wait, let's check with a reference. The measure of the angle formed by two tangents drawn from an external point to a circle is equal to the difference between 180° and the measure of the central angle between the two radii to the points of tangency. So if the central angle is θ, then the angle between tangents is 180° - θ. Now, in the diagram, the central angle θ is equal to 180° - 124° = 56°? No, that's not. Wait, the 124° angle is the central angle? No, the 124° angle is adjacent to the central angle. Wait, maybe the central angle is 180° - 124° = 56°, and then the angle between tangents is 180° - 56° = 124°? No, that's not right. Wait, let's take a simple case: if the central angle is 180° (a straight line), then the angle between tangents would be 0°, which makes sense (the tangents would be parallel). If the central angle is 90°, then the angle between tangents is 90°, which is a right angle. Wait, no, that formula is wrong. Wait, the correct formula is: the measure of the angle between two tangents drawn from an external point to a circle is equal to 180° minus the measure of the central angle between the two radii. So if the central angle is θ, then the angle between tangents is 180° - θ. Wait, no, let's derive it. The two radii are perpendicular to the tangents, so we have two right angles (90° each) at the points of tangency. The quadrilateral has four angles: 90°, 90°, θ (central angle), and x (angle between tangents). Sum of angles in quadrilateral: 90 + 90 + θ + x = 360 ⇒ θ + x = 180 ⇒ x = 180 - θ. So that's correct. Now, what is θ? The central angle between the two radii. In the diagram, the given angle is 124°, and since θ and 124° are supplementary (they form a linear pair, i.e., they are on a straight line), so θ = 180° - 124° = 56°. Then x = 180° - 56° = 124°? No, that can't be. Wait, no, the 124° angle is the central angle? Wait, maybe I misread the diagram. If the 124° angle is the central angle (θ), then x = 180° - 124° = 56°. Ah! That's probably the mistake. I thought the 124° angle was adjacent to θ, but actually, the 124° angle is θ, the central angle. Wait, let's look at the diagram again: O is the center, there is a line from O to a point on the circle, and another line from O to the other point on the circle, forming a 124° angle at O. So that 124° is the central angle (θ). Then, using the formula x = 180° - θ, so x = 180° - 124° = 56°. Yes, that makes sense. Because if the central angle is 124°, then the angle between the tangents is 180° - 124° = 56°. Let's verify: two radii, angle between them 124°, two tangents, each perpendicular to a radius (90°), so quadrilateral angles: 90°, 90°, 124°, and x. Sum: 90 + 90 + 124 + x = 360 ⇒ 304 + x = 360 ⇒ x = 56°. Yes! That's correct. I had the central angle wrong earlier. So the central angle θ is 124°, so x = 180° - 124° = 56°.

Step2: Calculate x

Using the formula for the angle between two tangents: \( x = 180^\circ - \theta \), where \( \theta = 124^\circ \) (the central angle).

\( x = 180^\circ - 124^\circ = 56^\circ \)

Answer:

\( 56 \)