Select The Correct Answers From Each Drop-down Menu To Complete The Steps In The Proof That Show Quadrilateral KITE, With Vertices K ( 0 , − 2 K (0, -2 K ( 0 , − 2 ], I ( 1 , 2 I (1, 2 I ( 1 , 2 ], T ( 7 , 5 T (7, 5 T ( 7 , 5 ], And E ( 4 , − 1 E (4, -1 E ( 4 , − 1 ], Is A Kite.Using The Distance
A kite is a type of quadrilateral that has two pairs of adjacent sides with equal lengths. In this article, we will explore the properties of a kite and use the distance formula to prove that the given quadrilateral KITE is indeed a kite.
What is a Kite in Geometry?
A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. The two pairs of adjacent sides are also known as the diagonals of the kite. The diagonals of a kite intersect at a right angle, and the two pairs of adjacent sides are equal in length.
Properties of a Kite
The properties of a kite include:
- Two pairs of adjacent sides with equal lengths
- Diagonals that intersect at a right angle
- Two pairs of congruent triangles
Using the Distance Formula to Prove KITE is a Kite
To prove that the quadrilateral KITE is a kite, we need to use the distance formula to show that the two pairs of adjacent sides are equal in length.
Step 1: Find the Length of the Sides of KITE
To find the length of the sides of KITE, we need to use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Step 2: Find the Length of the Diagonals of KITE
To find the length of the diagonals of KITE, we need to use the distance formula. The diagonals of a kite are the lines that connect the vertices of the kite.
Step 3: Prove that the Two Pairs of Adjacent Sides are Equal in Length
To prove that the two pairs of adjacent sides are equal in length, we need to use the distance formula to find the length of each side.
Step 4: Prove that the Diagonals of KITE Intersect at a Right Angle
To prove that the diagonals of KITE intersect at a right angle, we need to use the distance formula to find the length of each diagonal.
Step 5: Prove that the Two Pairs of Congruent Triangles are Equal in Length
To prove that the two pairs of congruent triangles are equal in length, we need to use the distance formula to find the length of each triangle.
The Proof
To prove that the quadrilateral KITE is a kite, we need to complete the following steps:
- Find the length of the sides of KITE
- Find the length of the diagonals of KITE
- Prove that the two pairs of adjacent sides are equal in length
- Prove that the diagonals of KITE intersect at a right angle
- Prove that the two pairs of congruent triangles are equal in length
Step 1: Find the Length of the Sides of KITE
To find the length of the sides of KITE, we need to use the distance formula.
| Side | Coordinates | Length |
|---|---|---|
| KI | (0, -2) - (1, 2) | √((1 - 0)^2 + (2 - (-2))^2) = √(1 + 16) = √17 |
| IT | (1, 2) - (7, 5) | √((7 - 1)^2 + (5 - 2)^2) = √(36 + 9) = √45 |
| TE | (7, 5) - (4, -1) | √((4 - 7)^2 + (-1 - 5)^2) = √(9 + 36) = √45 |
| EK | (4, -1) - (0, -2) | √((0 - 4)^2 + (-2 - (-1))^2) = √(16 + 1) = √17 |
Step 2: Find the Length of the Diagonals of KITE
To find the length of the diagonals of KITE, we need to use the distance formula.
| Diagonal | Coordinates | Length |
|---|---|---|
| KT | (0, -2) - (7, 5) | √((7 - 0)^2 + (5 - (-2))^2) = √(49 + 49) = √98 |
| IE | (1, 2) - (4, -1) | √((4 - 1)^2 + (-1 - 2)^2) = √(9 + 9) = √18 |
Step 3: Prove that the Two Pairs of Adjacent Sides are Equal in Length
To prove that the two pairs of adjacent sides are equal in length, we need to compare the lengths of the sides.
| Side | Length |
|---|---|
| KI | √17 |
| IT | √45 |
| TE | √45 |
| EK | √17 |
As we can see, the lengths of the sides KI and EK are equal, and the lengths of the sides IT and TE are equal.
Step 4: Prove that the Diagonals of KITE Intersect at a Right Angle
To prove that the diagonals of KITE intersect at a right angle, we need to use the distance formula to find the length of each diagonal.
| Diagonal | Length |
|---|---|
| KT | √98 |
| IE | √18 |
As we can see, the length of the diagonal KT is √98, and the length of the diagonal IE is √18. Since the length of the diagonal KT is greater than the length of the diagonal IE, we can conclude that the diagonals of KITE intersect at a right angle.
Step 5: Prove that the Two Pairs of Congruent Triangles are Equal in Length
To prove that the two pairs of congruent triangles are equal in length, we need to use the distance formula to find the length of each triangle.
| Triangle | Length |
|---|---|
| KIT | √(KI^2 + IT^2) = √(17 + 45) = √62 |
| TEK | √(TE^2 + EK^2) = √(45 + 17) = √62 |
As we can see, the lengths of the triangles KIT and TEK are equal.
Conclusion
In conclusion, we have proven that the quadrilateral KITE is a kite by showing that the two pairs of adjacent sides are equal in length, the diagonals of KITE intersect at a right angle, and the two pairs of congruent triangles are equal in length.
Final Answer
In this article, we will answer some of the most frequently asked questions about kites in geometry.
Q: What is a kite in geometry?
A: A kite is a type of quadrilateral that has two pairs of adjacent sides with equal lengths. The two pairs of adjacent sides are also known as the diagonals of the kite.
Q: What are the properties of a kite?
A: The properties of a kite include:
- Two pairs of adjacent sides with equal lengths
- Diagonals that intersect at a right angle
- Two pairs of congruent triangles
Q: How do you prove that a quadrilateral is a kite?
A: To prove that a quadrilateral is a kite, you need to show that the two pairs of adjacent sides are equal in length, the diagonals intersect at a right angle, and the two pairs of congruent triangles are equal in length.
Q: What is the difference between a kite and a rhombus?
A: A kite and a rhombus are both quadrilaterals with two pairs of adjacent sides with equal lengths. However, a kite has one pair of opposite angles that are equal, while a rhombus has all four angles that are equal.
Q: Can a kite have all sides equal?
A: Yes, a kite can have all sides equal. In this case, the kite is also a rhombus.
Q: Can a kite have all angles equal?
A: No, a kite cannot have all angles equal. A kite has one pair of opposite angles that are equal, while a rhombus has all four angles that are equal.
Q: What is the relationship between the diagonals of a kite?
A: The diagonals of a kite intersect at a right angle.
Q: Can the diagonals of a kite be equal in length?
A: Yes, the diagonals of a kite can be equal in length.
Q: What is the relationship between the sides and diagonals of a kite?
A: The sides and diagonals of a kite are related in such a way that the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of the sides.
Q: Can a kite be a regular polygon?
A: No, a kite cannot be a regular polygon. A kite has two pairs of adjacent sides with equal lengths, while a regular polygon has all sides equal.
Q: Can a kite be a square?
A: No, a kite cannot be a square. A kite has one pair of opposite angles that are equal, while a square has all four angles that are equal.
Q: Can a kite be a rectangle?
A: No, a kite cannot be a rectangle. A kite has two pairs of adjacent sides with equal lengths, while a rectangle has all sides equal.
Conclusion
In conclusion, we have answered some of the most frequently asked questions about kites in geometry. We hope that this article has provided you with a better understanding of the properties and characteristics of kites.
Final Answer
The final answer is: