*tag-

The core idea of this root was:

touching, handling

This theme also at the core of the Latin offspring tangere, which meant to touch. And also in taxare, which had an additional theme of “gauging” the value. Think of someone touching an object to inspect it.
Here’s a little (incomplete) overview over English family members, many of which are from Latin:

• tactile (“touch-related”)
• tangent (“touching”)
• tangible (“touchable”)
• contact (“touch together”)
• attain (“attacking, getting to touch”)
• tactic (“how you arrange”)
• tax (“the estimate”)
• task (Originally, it was an assessement, but it also makes sense to think of it in terms of touching)
• intact (originally: “not touched”)
• entire (same thing pretty much)
• integer, integrity
• integrate (“make whole, make untouched”)