Question

Is the shape a kite? (1) The diagonals are perpendicular (2) Two pairs of sides are equal A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. \(\quad 1\) and 2 together are not sufficient

Short answer

Option C: 1 and 2 together are needed to determine the shape is a kite.

Step-by-step solution

Understand the properties of a kite

A kite in geometry is a quadrilateral with two distinct pairs of adjacent sides that are equal. Additionally, one important property of a kite is that its diagonals are perpendicular to each other.

Analyze Statement 1

Statement 1 states that the diagonals are perpendicular. This property is necessary for a kite, but it is not sufficient to determine if a quadrilateral is a kite by itself. Other quadrilaterals, such as rhombuses, also have perpendicular diagonals.

Analyze Statement 2

Statement 2 states that two pairs of sides are equal. This is another characteristic of a kite. However, this condition alone is also found in other figures such as rhombuses and rectangles with equal sides in pairs, and cannot confirm the figure is a kite without further information.

Combine both statements

If both statements 1 and 2 are together, we have enough information to conclude that the quadrilateral is a kite. The combination of equal pairs of adjacent sides and perpendicular diagonals guarantees the figure must be a kite.

Choose the correct answer

Given the analysis, both statements together are necessary to determine that the given shape is a kite. Thus, the correct answer is option C: 1 and 2 together (need both).

Key concepts

kite properties

A kite is a special type of quadrilateral. To visualize, imagine a flying kite with a string. In geometry, a kite has specific properties:

• Two pairs of adjacent (next to each other) sides are equal in length.
• The diagonals intersect at a right angle (90 degrees).

These properties make kites unique compared to other quadrilaterals. Understanding these characteristics can help you identify and solve problems revolving around kites easily.

quadrilateral diagonals

In quadrilaterals, diagonals play a significant role in defining shapes. By definition, diagonals are line segments joining two opposite vertices. For example, in a kite:

• The diagonals are perpendicular, meaning they intersect at a 90-degree angle.
• One diagonal is usually the axis of symmetry, while the other divides the kite into two congruent triangles.

These properties help to distinguish kites from other shapes like rhombuses or rectangles, where diagonals may behave differently.

equal side pairs

Equal side pairs are a fundamental characteristic of kites. This means that in a kite, each pair of adjacent sides has the same length. Let's break this down:

• Pair 1: Side A = Side B
• Pair 2: Side C = Side D

Here, each pair is distinct and sets the kite apart. While other shapes, such as rectangles, also have equal pairs, what makes a kite unique is that the pairs are adjacent, not opposite. This can help confirm the identity of a kite when solving geometry problems.