Exercise 3 Page 623

In a previous solution, we studied how the area of a kite changes when the length of one or both of the diagonals is doubled.

The two conclusions we have drawn in that exercise were the following.

• If the length of one diagonal is doubled, the area is also doubled. To check this, we substitute 2d_1 for d_1 in the formula for the area of a kite.

  A_2 = 1/2(2 d_1) d_2 ⇒ A_2 &= 2(1/2 d_1 d_2) ⇒ A_2 &= 2A_1 ✓The same thing happens when we substitute 2d_2 for d_2 in the formula.A_2 = 1/2 d_1(2 d_2) ⇒ A_2 &= 2(1/2 d_1 d_2) ⇒ A_2 &= 2A_1 ✓

  • If the length of both diagonals is doubled, the area is quadrupled. To check this, we substitute 2d_1 for d_1 and 2d_2 for d_2 in the formula for the area of a kite.A_3 = 1/2(2 d_1)(2 d_2) ⇒ A_3 &= 4(1/2 d_1 d_2) ⇒ A_3 &= 4A_1 ✓

  In this exercise, we will write a proof of each of those facts.

  Two-Column Proofs

  In the first two-column proof, we will prove that when the length of one diagonal is doubled, the area of the kite is also doubled. First, we will consider the situation where the diagonal of length d_1 is doubled.

  Statements	Reasons
  1. ABCD is a kite with area A_1 = 12d_1d_2, where d_1 and d_2 are the length of the diagonals.	1. Given
  2. Let PQRS be a kite with diagonals x=2d_1 and y=d_2.	2. Construction
  3. A_(PQRS) = 1/2xy	3. Area of a kite
  4. A_(PQRS) = 1/2* 2d_1d_2	4. Substitution
  5. A_(PQRS) = 2(1/2d_1d_2)	5. Commutative Property of Multiplication
  6. A_(PQRS) = 2A_1	6. Substitution

  Now, let's write a proof for the situation when the diagonal with length d_2 is doubled.

  Statements	Reasons
  1. ABCD is a kite with area A_1 = 12d_1d_2, where d_1 and d_2 are the length of the diagonals.	1. Given
  2. Let PQRS be a kite with diagonals x=d_1 and y=2d_2.	2. Construction
  3. A_(PQRS) = 1/2xy	3. Area of a kite
  4. A_(PQRS) = 1/2* d_12d_2	4. Substitution
  5. A_(PQRS) = 2(1/2d_1d_2)	5. Commutative Property of Multiplication
  6. A_(PQRS) = 2A_1	6. Substitution

  In the next two-column proof, we show that if the length of both diagonals is doubled, the area of the kite is quadrupled.

  Statements	Reasons
  1. ABCD is a kite with area A_1 = 12d_1d_2, where d_1 and d_2 are the length of the diagonals.	1. Given
  2. Let WZYZ a kite with diagonals j=2d_1 and k=2d_2.	2. Construction
  3. A_(WZYZ) = 1/2jk	3. Area of a kite
  4. A_(WZYZ) = 1/2* 2d_1* 2d_2	4. Substitution
  5. A_(WZYZ) = 4(1/2d_1d_2)	5. Commutative Property of Multiplication
  6. A_(WZYZ) = 4A_1	6. Substitution