Question

1. true or false: ab is tangent to circle c.

29
a 21 b
10
d
c
false
true

Explanation:

Step1: Recall Tangent-Segment Property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, if \( AB \) is tangent to circle \( C \) at \( B \), then \( \angle ABC = 90^\circ \), and triangle \( ABD \) (wait, actually triangle \( ABC \)? Wait, \( CB \) is radius, length \( 10 \), \( AB = 21 \), \( AD = 29 \). Let's check if triangle \( ABD \) is right-angled at \( B \).

Step2: Apply Pythagorean Theorem

Calculate \( AB^2 + BD^2 \). \( BD = BC + CD \), but \( BC = 10 \), \( CD \) is also radius, so \( BD = 10 + 10 = 20 \)? Wait, no, \( C \) is the center, so \( CB \) is radius (length 10), \( CD \) is also radius (length 10), so \( BD = BC + CD = 10 + 10 = 20 \). Then \( AB = 21 \), \( BD = 20 \), \( AD = 29 \).

Check \( AB^2 + BD^2 = 21^2 + 20^2 = 441 + 400 = 841 \). And \( AD^2 = 29^2 = 841 \). So \( AB^2 + BD^2 = AD^2 \), which means \( \angle ABD = 90^\circ \). Since \( CB \) is radius and \( AB \perp BD \) (and \( BD \) is a straight line with \( CB \) as part of it, so \( AB \perp CB \)). Therefore, \( AB \) is perpendicular to the radius \( CB \) at \( B \), so \( AB \) is tangent to circle \( C \).

Answer:

True