Question

find the length of lk

Explanation:

Step1: Identify Tangent and Radius

In the circle, \( LJ \) is a radius (length \( 7 \)) and \( KJ \) is a tangent to the circle at \( J \), so \( \angle LJK = 90^\circ \) (tangent is perpendicular to radius). Let \( LK = x \), then \( KJ = 19 \), \( LJ = 7 \), and by Pythagorean theorem: \( KJ^2 + LJ^2 = LK^2 \)? Wait, no—wait, actually, if \( KJ \) is tangent, then \( KJ \perp LJ \), so triangle \( LJK \) is right-angled at \( J \). Wait, but maybe \( LK \) is a secant? Wait, no, the diagram: \( KJ \) is tangent (length 19), \( LJ \) is radius (7), and \( LK \) is a line from \( K \) to center \( L \)? Wait, no, maybe the center is \( L \)? Wait, the circle has center \( L \), so \( LJ \) is radius (7), \( KJ \) is tangent, so \( LJ \perp KJ \). Then \( KJ = 19 \), \( LJ = 7 \), so \( LK \) is the hypotenuse of right triangle \( LJK \). Wait, but that would be \( LK = \sqrt{19^2 + 7^2} \)? Wait, no, that can't be. Wait, maybe the tangent-secant theorem? Wait, no, if \( KJ \) is tangent and \( KI \) is secant (but the diagram shows \( LK \) and \( LJ \)). Wait, maybe I misread. Wait, the problem is to find \( LK \), with \( KJ = 19 \), \( LJ = 7 \), and \( LJ \perp KJ \). Then by Pythagoras: \( LK = \sqrt{KJ^2 + LJ^2} \)? Wait, \( 19^2 = 361 \), \( 7^2 = 49 \), sum is \( 410 \), square root of \( 410 \approx 20.25 \). Ah, that matches one of the options (20.25).

Step2: Apply Pythagorean Theorem

In right triangle \( LJK \) (right-angled at \( J \)):
\( LK^2 = KJ^2 + LJ^2 \)
Substitute \( KJ = 19 \), \( LJ = 7 \):
\( LK^2 = 19^2 + 7^2 = 361 + 49 = 410 \)
\( LK = \sqrt{410} \approx 20.25 \)

Answer:

20.25