Deltoid Practice Problems: Kite Properties & Solutions

First, let's recall the formula for the area of a rhombus:

(Diagonal 1 * Diagonal 2) divided by 2

Now we will substitute the known data into the formula, giving us the answer:

(12*16)/2
192/2=
96

First, we recall the formula for the area of a kite: multiply the lengths of the diagonals by each other and divide the product by 2.

We substitute the known data into the formula:

$\frac{8\cdot DB}{2}=32$

We reduce the 8 and the 2:

$4DB=32$

Divide by 4

$DB=8$

Initially, let us examine the basic properties of a deltoid (or kite):

A quadrilateral is classified as a deltoid if:

• It has two distinct pairs of adjacent sides that are equal in length.

In the question's image, we observe the following:

• There are lines connecting A to B, B to C, C to D, and D to A, suggesting a typical quadrilateral.
• The shape, given its central symmetry (as it is formed by joining these particular points which extend equal lines), is reminiscent of a symmetric or bilaterally mirrored formation.
• Given the symmetry, it suggests all internal angles are less than 180 degrees, confirming the figure as a convex shape.

From this analysis, the quadrilateral satisfies the characteristic of having pairs of equal adjacent sides which confirms it as a deltoid. The symmetry suggests it is not concave (which occurs when at least one interior angle is greater than 180 degrees).

Therefore, the given quadrilateral, based on its properties and symmetry, is a convex deltoid.

In order to answer the question we must recall some properties of the deltoid. For this purpose, let's draw the deltoid $ABCD$ where we connect every two non-adjacent vertices (meaning - draw the diagonals) and mark the intersection of the diagonals with the letter $E$:

$$

Let's recall two properties of the deltoid that will help us answer the question ( from the previous drawing):

a. Definition of a deltoid - A deltoid is a convex quadrilateral with two pairs of adjacent equal sides:

$BA=BC\\ DA=DC$

b. The diagonals in a deltoid are perpendicular to each other:

$AC\perp BD\\ \updownarrow\\ \sphericalangle BEA= \sphericalangle AED= \sphericalangle DEC= \sphericalangle CEB=90\degree$

Now we can clearly answer the question that was asked, and the answer is that the deltoid can indeed be described as composed of two isosceles triangles since triangles: $\triangle ABC,\hspace{6pt}\triangle ADC$ are isosceles - (from property a' mentioned earlier):

Or can be described as composed of four right triangles, since triangles: $\triangle AEB,\hspace{6pt}\triangle CEB,\hspace{6pt}\triangle AED,\hspace{6pt}\triangle CED$ are right triangles (from property b' mentioned earlier):

Therefore, the correct answer is answer a'.

Let's begin by reminding ourselves of the formula for the area of a kite

$\frac{Diagonal1\times Diagonal2}{2}$

Both these values are given to us in the figure thus we can insert them directly into the formula:

(4*7)/2

28/2

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