The Deltoid and Everything You Need to Know to Verify It
What is a Kite or Deltoid?
In geometry, a deltoid is defined as a quadrilateral consisting of 2isosceles triangles that share a common base.
So, what is the identification of a deltoid in the family of quadrilaterals?
A quadrilateral that has 2 pairs of equal adjacent sides
Example:
If given : AD=AB,DC=BC
Then:ABCD is a deltoid by definition.
2isosceles triangles with a common base form a deltoid.
The sum of the angles in the deltoid is360° degrees.
The area of the deltoid contains the number of quadrilaterals that cover the selected parts of the plane.
The perimeter of the deltoid is the length of the thread with which we border the outline of the deltoid and is measured in units of length in meters or cm.
Some Basic Concepts of the Kite
Main diagonal: The diagonal that passes between the identical sides in a kite.
Secondary diagonal: The common base of 2 isosceles triangles in a kite is called the secondary diagonal.
Vertex angles: The angles between the equal sides in a kite.
Base angles: The angles through which the common base passes.
Types of Kites
Convex Kite
Convex Kite: A kite with diagonals on the inside (as in the images of the kites above)
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Concave Kite
Concave Kite: A kite with one of its diagonals outside (like a kind of bowl).
On many occasions, when we sit on the beach facing the sea, we observe a good number of kites. Have you looked at their shape? This is a deltoid shape. The deltoid has a somewhat complicated form. It's a quadrilateral but not a square, and it has a shape similar to a rhombus and a parallelogram, but their definitions are different. In this article, we will learn what a deltoid is and how to identify it.
Who Else Belongs to the Kite Family?
Diamond Shape
Rhombus: All sides are equal vertical diagonals, diagonals that cross each other and bisect the angles, from each side we look at the quadrilateral of the kite. The rhombus is actually an equilateral kite.
Square
Square: The most elaborate of the group: its diagonals are perpendicular and intersect; they cross the angles as in a rhombus, but in a square, the lengths of the diagonals are equal as in a rectangle. Also, from every side we look, we'll notice 2 isosceles triangles with a common base, so the characteristics of the kite will also be present in it. The square is a kite with equal sides and angles (all angles are right angles).
And, of course, the deltoid itself:
2 pairs of equal adjacent sides.
Deltoid Test
Why are the base angles equal in a kite?
We will use the definition of a Kite: 2 equilateral triangles with a common base
Therefore:AD=AB, and also CD=CB.
According to this:∢ABD=∢ADB Because the base angles in an equilateral triangle are equal
Also:∢BDC=∢DBC Base angles in an isosceles triangle are equal
Therefore:∢ABC=∢ADC We combine equal angles with equal angles so that the sum of the angles is equal (the total amount)
Even if we overlaid the triangles: △ABC with △ADC
We would obtain:
AB=AD (given)
BC=DC (given)
AC=AC (common side)
Therefore, we can conclude:
△ABC≅△ADC (according to the superposition theorem: side, side, side)
∢ABC=∢ADC (Corresponding angles in equal overlaid triangles)
As a result of the overlay, the kite principle can be deduced:
The main diagonal in the kite intersects the angles, crosses a secondary diagonal, and is perpendicular to it.
△ABC≅△ADC (according to the superposition theorem: side, side, side) Proven
Therefore:∢DAC=∢BAC
Also:∢BCA=∢DCA, Corresponding angles in equal overlaid triangles
The main diagonal in the kite intersects a secondary diagonal and is perpendicular to it.
According to the data:AD=AB After all, triangle ADB is an isosceles triangle.
In an isosceles triangle the vertex angle is perpendicular to the base and bisects it.
Therefore:AC⊥DB and also: DM=BM
From this, we can calculate the missing sides and the missing angles in the given kite:
ABCD is a kite,
FindX,Y,α,β in the given kite
X=AB=AD
X=5cm
According to the definition of a kite.
∢BAC=α=40° The main diagonal of the kite intersects the angles.
∢ACD=β=50° The main diagonal of the kite intersects the angles.
Y=3cm, the main diagonal in the kite intersects the secondary diagonal.
Calculating the perimeter of a kite is done by adding up all its sides:
5+5+4+4=18cm
And the calculation of the area of the deltoid is done using the product of the diagonals divided by two:
Calculation of the secondary diagonal:6cm=3+3=BD
And to calculate the length of the main diagonal ACwe use the Pythagorean theorem in right-angled triangles formed by the diagonals (as it has been proven to us that they are perpendicular to each other)
And therefore, in the triangle△ABO we obtain:
AO2+32=52
AO2+9=25
AO2=16 and we apply the
AO=4cm
And in the triangle△CBO we obtain:
CO2+32=42
9+CO2=16
CO2=7
2.645cm=CO
Therefore, the length of the main diagonal is equal to:
4+2.645=6.645cm
We can calculate the area of the deltoid:
26.645×6=19.935cm2
Deltoid Test: What is the necessary condition to get a deltoid?
Does every quadrilateral whose diagonals are perpendicular form a kite?
The answer is: not necessarily
See example:
If that's the case, what is the additional condition for the vertical diagonals that a kite requires for acceptance?
Let's check, here we have a quadrilateral where one diagonal crosses the other and is perpendicular to it, is it necessarily accepted to be a kite?
Given:
DO=BO
AC⊥DB
Is it accepted as a kite?
Since DO=BO and also AC⊥DB
Therefore, we can conclude that AD=AB and also DC=BC (in a triangle where the altitude is also the median, it is an isosceles triangle)
According to this, ABCD is a kite according to the definition: 2 isosceles triangles on a common base form a kite.
Another condition for a quadrilateral to be a kite: one of the diagonals bisects the angles.
Given:
∢A1=∢A2,∢C1=∢C2
Prove:ABCD is a kite
Proof:
∢A1=∢A2 (Given)
∢C1=∢C2(Given)
AC=AC (common side)
Therefore:
△ABC≅△ADC (by the Angle-Side-Angle postulate)
Therefore:
AB=AD
BC=DC (corresponding sides in congruent triangles are equal)
If you're interested in learning how to calculate areas of other geometric shapes, you can check out one of the following articles:
How do you calculate the area of a trapezoid?
How to calculate the area of a triangle
The area of a parallelogram: what is it and how is it calculated?
Circular area
Surface area of triangular prisms
How do you calculate the area of a rhombus?
How to calculate the area of a regular hexagon?
How to calculate the area of an orthohedron
Proof by contradiction
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Exercises
Exercise 1
In the following exercise, it is necessary to know the Pythagorean Theorem.
Given: The deltoid ABCD(AB=AD,DC=BC).
The diagonal of the deltoid
Crosses point o.
Task
Calculate the side of CD.
Solution:
Data: Deltoid ABCD.
Data: Area of ABCD=48cm2.
Data: OC=4.
Data: AO=12.
Sum of the parts AC=AO+OC.
Solving for variables: AC=12+4.
Calculation: AC=16.
Formula to calculate the area of deltoid abcd= (AC×BD):2.
Solving for variables: 216×BD=48.
Calculation: 96=16×BD.
BD=6cm.
In the deltoid, the main diagonal (AC) crosses the secondary diagonal (BD)OB=DO.
Sum of the parts BD=DO+OB.
Solving for variables DO+OB=6.
DO2=6.
DO=3.
AC⊥BD In the deltoid the diagonals are perpendicular to each other.
Between perpendicular lines there are right angles (90° degrees).
Let's look at the right triangle \( COD \):
DO2+CO2=CD2 Pythagorean Theorem.
DO=3,CO=4 Proof.
32+42=CD2 Solving for variable.
9+16=CD2.
CD2=25.
CD=5cm Q.E.D.
Answer:
CD=5cm.
Exercise 2
Given: The deltoid ABCD
The area of the deltoid is equal to 6a
The main diagonal is equal to 2a+2
The secondary diagonal is equal to a
Task
Calculate the value of:a
Solution:
Given the area of the deltoid:
A=2AC×DB=16a
AC=2a+2 (Main diagonal)
DB=a (Secondary diagonal)
12a=(2a+2)a (Expanding the parenthesis)
2a2+2a=12a
2a2=10a /: (Divide by a)
2a=10 /: (Divide by 2)
a=5cm
Answer:
a=5cm
Exercise 3
Given a kite ABCD
The diagonal DB is equal to 5cm.
The side AD is equal to 4cm
Homework
Is it possible to calculate the area of the kite? If so, calculate its area.
Solution
Formula for calculating the area of the kite
A=2AC×DB
Given that DB=5cm
AD=4cm
The formula cannot be applied because the diagonal AC is not given, and there is no information provided that would help to find it.
Answer
It is not possible to calculate the area of the kite
Exercise 4
Given the kite ABCD
Given that Area ABCD=42cm2
Given that BD=14
Task:
Calculate the value of AO
Solution:
Given that ABCD is a kite
The area of the kite ABCD is equal to 42cm
BD=14
The formula to calculate the area of the kite is:
2AC×BD= A
2AC×14= 42
84=14×AC
AC=6
In the kite, the main diagonal crosses the secondary diagonal. AO=OC
OC+AO=AC (Sum of the parts)
OC+AO=AC variable isolation (AC=6,OC=AO)
2AO=6 Q.E.D
AO=3cm
Answer:
AO=3cm
Exercise 5
Given that the deltoid ABCD is enclosed within a rectangle KMNH
Side AC=8
Height DO of the triangle ADC is equal to 3cm
Task:
Calculate the value of the white area
Solution:
Given that AC=8cm
Given that DO=3cm
To calculate the dotted area, we calculate the area of the rectangle and subtract the area of the deltoid.
We start with the area of the rectangle:
A= MN×KM
MN=AC=8 (in an equilateral parallel rectangle)
DO=OB=3
(The main diagonal in a rhombus is perpendicular to the secondary diagonal and crosses it)
Therefore: DO+OB=DB=6
DB=KM (Equal parallel sides in a rectangle)
Area of the rectangle:A=6×8=48cm2
Area of the deltoid:
A=ABCD=2AC×DB=28×6=24cm2
Area of rectangle - Area of deltoid = Dotted area
48−24=24cm2
Answer:
The answer is 24cm2
Exercise 6
Given the concave kite ABCD
Given that the diagonal AC is equal to 75% of the diagonal DB
The area of the kite is equal to 108Xcm2.
Task:
Calculate the side DB
DB=X
Solution:
Given the area of the kite =108X
Given: DB=X
Given:
AC=X75%=43X
This is because AC is equal to 75% of DB which is equal to 43,DB