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Chapter 11: Problem 38

What happens to the area of a kite if you double the length of one of the diagonals? if you double the length of both diagonals? Justify your answer.

Short Answer

Expert verified
If one of the diagonals of a kite is doubled, the area of the kite will double. If both diagonals are doubled, the area will quadruple.

Step by step solution

01

Understand the Concept of Kite

A kite is a quadrilateral whose two pairs of adjacent sides are equal. The area of the kite can be calculated by multiplying the lengths of the two diagonals and then dividing by 2; Area = 1/2 * d1 * d2 where d1 and d2 are the lengths of the two diagonals.
02

Determine the Effect of Doubling One Diagonal

If one of the diagonals is doubled, then the new area becomes Area = 1/2 * 2d鈧� * d鈧� or Area = d鈧� * d鈧�. Thus, doubling one of the diagonals causes the area to double.
03

Determine the Effect of Doubling Both Diagonals

If both the diagonals are doubled, then the new area is Area = 1/2 * 2d鈧� * 2d鈧� or Area = 2d鈧� * d鈧�. Thus, doubling both diagonals causes the area to quadruple.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area of a Kite
The area of a kite is a fascinating concept due to its unique shape and properties. In geometry, a kite is a type of quadrilateral, but we will delve more deeply into that in another section. When it comes to calculating the area of a kite, it's essential to focus on the diagonals. The formula for finding the area relies on these diagonals, which may differ in length. It is given by \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]. This formula captures the distinct aspect of kites since the area is directly influenced by the lengths of both diagonals. If you change the length of one or both diagonals, the area will change correspondingly.
  • Doubling one diagonal: The area doubles because you're multiplying by twice its original length.
  • Doubling both diagonals: The area becomes four times larger, since you're effectively multiplying by four.
Quadrilaterals
Kites belong to the family of quadrilaterals, which are four-sided polygons. Quadrilaterals are classified based on their sides, angles, and symmetries. Common examples include squares, rectangles, and rhombuses. The distinction between these figures is based on their lines, angles, and side relationships. For kites:
  • They have two pairs of adjacent sides that are equal.
  • The opposite angles, where the unequal sides meet, are equal.
Their distinct shape makes them versatile in problem-solving within geometry, particularly concerning area calculations as previously discussed.
Diagonal Properties
Diagonals play a crucial role in the properties and calculations of many quadrilaterals, and the kite is no exception. A diagonal in a quadrilateral is a line segment connecting non-adjacent vertices. In a kite:
  • Diagonals intersect each other at right angles, forming perpendicular lines.
  • One diagonal acts as an axis of symmetry, dividing the kite into two congruent triangles.
  • The diagonals are not generally equal, which contrasts with a rectangle where they are equal.
Understanding the properties of diagonals helps in leveraging formulas for area and understanding the kite's geometric behavior.
Kite Properties
A kite, with its unique geometric form, has several properties that make it distinct. These properties define its shape and behavior in geometric problems. Key properties include:
  • The kite has exactly one pair of opposite angles that are equal.
  • Diagonals intersect at 90 degrees, ensuring the formation of right angles at the center.
  • Two pairs of adjacent sides are equal, which provides its distinctive shape.
These properties are instrumental, especially when answering questions about area, angles, and symmetry. In problems requiring the manipulation of dimensions, such as doubling diagonals, understanding these properties becomes crucial in visualizing and calculating the outcomes.

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