Given triangle ABC right at A, AH is height . Draw \(HD\perp AB\) at D, \(HE\perp AC\) at E

Prove that : AH² = BH.HC

                    AB² = BH.BC 

Alicia is framing a rectangular photo that measures 8 inches high by 12 inches wide. However, the frame she has was designed to hold a picture with sides in the ratio 4:5. If the frame Alicia has is the smallest possible frame that holds the photo, how much extra area will there be? Express your answer as a decimal to the nearest tenth.

In how many ways can the integers 1 through 6 be written horizontally in a row so that the sum of any two adjacent integers is odd?

What is the minimum number of people that must be in a room to guarantee that at least one person has a birthday whose day is a single digit, assuming that no two people have the same birthday?

To cater a party, Fancy Caterers charges a certain amount per guest plus a fixed delivery fee. The total cost for a party with 25 guests is twice the cost of a party with 10 guests. If the cost of a party with n guests is twice the cost of a party with 25 guests, what is the value of n?