diff --git a/Library/Homebrew/PATH.rb b/Library/Homebrew/PATH.rb
index f9c347d6e8..20b83d0d36 100644
--- a/Library/Homebrew/PATH.rb
+++ b/Library/Homebrew/PATH.rb
@@ -1,52 +1,71 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 # Represention of a `*PATH` environment variable.
 #
 # @api private
 class PATH
+  extend T::Sig
+
   include Enumerable
   extend Forwardable
 
   def_delegator :@paths, :each
 
+  # FIXME: Enable cop again when https://github.com/sorbet/sorbet/issues/3532 is fixed.
+  # rubocop:disable Style/MutableConstant
+  Element = T.type_alias { T.nilable(T.any(Pathname, String, PATH)) }
+  private_constant :Element
+  Elements = T.type_alias { T.any(Element, T::Array[Element]) }
+  private_constant :Elements
+  # rubocop:enable Style/MutableConstant
+
+  sig { params(paths: Elements).void }
   def initialize(*paths)
-    @paths = parse(*paths)
+    @paths = parse(paths)
   end
 
+  sig { params(paths: Elements).returns(T.self_type) }
   def prepend(*paths)
-    @paths = parse(*paths, *@paths)
+    @paths = parse(paths + @paths)
     self
   end
 
+  sig { params(paths: Elements).returns(T.self_type) }
   def append(*paths)
-    @paths = parse(*@paths, *paths)
+    @paths = parse(@paths + paths)
     self
   end
 
+  sig { params(index: Integer, paths: Elements).returns(T.self_type) }
   def insert(index, *paths)
-    @paths = parse(*@paths.insert(index, *paths))
+    @paths = parse(@paths.insert(index, *paths))
     self
   end
 
+  sig { params(block: T.proc.params(arg0: String).returns(T::Boolean)).returns(T.self_type) }
   def select(&block)
     self.class.new(@paths.select(&block))
   end
 
+  sig { params(block: T.proc.params(arg0: String).returns(T::Boolean)).returns(T.self_type) }
   def reject(&block)
     self.class.new(@paths.reject(&block))
   end
 
+  sig { returns(T::Array[String]) }
   def to_ary
     @paths.dup.to_ary
   end
   alias to_a to_ary
 
+  sig { returns(String) }
   def to_str
     @paths.join(File::PATH_SEPARATOR)
   end
   alias to_s to_str
 
+  sig { params(other: T.untyped).returns(T::Boolean) }
   def ==(other)
     if other.respond_to?(:to_ary) && to_ary == other.to_ary
       true
@@ -57,10 +76,12 @@ class PATH
     end
   end
 
+  sig { returns(T::Boolean) }
   def empty?
     @paths.empty?
   end
 
+  sig { returns(T.nilable(T.self_type)) }
   def existing
     existing_path = select(&File.method(:directory?))
     # return nil instead of empty PATH, to unset environment variables
@@ -69,10 +90,11 @@ class PATH
 
   private
 
-  def parse(*paths)
+  sig { params(paths: T::Array[Elements]).returns(T::Array[String]) }
+  def parse(paths)
     paths.flatten
          .compact
-         .flat_map { |p| Pathname.new(p).to_path.split(File::PATH_SEPARATOR) }
+         .flat_map { |p| Pathname(p).to_path.split(File::PATH_SEPARATOR) }
          .uniq
   end
 end