Segment AB is a diameter. Segments AB and CD are perpendicular. Find the measure of arc BD if the measure of arcAC is 43. 133 47 137 86

Given the problem, we need to find the measure of arc \( BD \) given that segment \( AB \) is a diameter and segments \( AB \) and \( CD \) are perpendicular. Additionally, we know that the measure of arc \( AC \) is 43 degrees.<br /><br />1. **Understanding the Geometry:**<br /> - Since \( AB \) is a diameter, it divides the circle into two equal semicircles, each measuring 180 degrees.<br /> - Segments \( AB \) and \( CD \) being perpendicular implies that \( CD \) is a chord that intersects the circle at right angles to the diameter \( AB \).<br /><br />2. **Arc Calculation:**<br /> - The measure of arc \( AC \) is given as 43 degrees.<br /> - Since \( AB \) is a diameter, the measure of arc \( ABC \) (which includes arc \( AC \) and arc \( CB \)) must be 180 degrees (as it is a semicircle).<br /><br />3. **Finding Arc \( CB \):**<br /> - To find the measure of arc \( CB \), we subtract the measure of arc \( AC \) from 180 degrees:<br /> \[<br /> \text{Measure of arc } CB = 180^\circ - 43^\circ = 137^\circ<br /> \]<br /><br />4. **Perpendicular Chord Property:**<br /> - Since \( CD \) is perpendicular to \( AB \), it implies that \( D \) lies directly opposite to \( C \) on the circle, forming a right angle at the center.<br /> - Therefore, the arc \( BD \) is the same as the arc \( CB \).<br /><br />5. **Conclusion:**<br /> - The measure of arc \( BD \) is equal to the measure of arc \( CB \), which we calculated to be 137 degrees.