SCALENE TRAPEZOID: PROPERTIES, FORMULAS AND EQUATIONS, EXAMPLES - MATHS - 2024

A scalene trapezoid is a polygon with four sides, two of which are parallel to each other, and with its four interior angles of different measures.

The quadrilateral ABCD is shown below, where sides AB and DC are parallel to each other. This is enough for it to be a trapezoid, but also, the interior angles α, β, γ and δ are all different, therefore the trapezoid is scalene.

Figure 1. Quadrilateral ABCD is trapezoid by condition 1 and scalene by condition 2. Source: F. Zapata.

Elements of the scalene trapezium

Here are the most characteristic elements:

-Bases and sides: the parallel sides of the trapezoid are its bases and the two non-parallel sides are the sides.

In a scalene trapezoid the bases are of different lengths and the lateral ones as well. However, a scalene trapezoid can have a lateral equal in length to a base.

-Median: is the segment that joins the midpoints of the lateral ones.

-Diagonals: the diagonal of a trapezoid is the segment that joins two opposite vertices. A trapezoid, like every quadrilateral, has two diagonals. In the scalene trapezoid they are of different length.

Other trapezoids

Besides the scalene trapezoid, there are other particular trapezoids: the right trapezoid and the isosceles trapezoid.

A trapezoid is a rectangle when one of its angles is right, while an isosceles trapezoid has its sides of equal length.

The trapezoidal shape has numerous applications at the design and industry level, such as in the configuration of aircraft wings, the shape of everyday objects such as tables, chair backs, packaging, purses, textile prints and more.

Figure 2. The trapezoidal shape is common in the wing configuration of airplanes. Source: Wikimedia Commons.

Properties

The properties of the scalene trapezoid are listed below, many of which extend to the other types of trapezoid. In what follows, when "trapezoid" is spoken of, the property will apply to any type, including scalene.

1. The median of the trapezoid, that is, the segment that joins the midpoints of its non-parallel sides, is parallel to any of the bases.

2.- The median of a trapezoid has a length that is the semisum of that of its bases and cuts its diagonals at the midpoint.

3.- The diagonals of a trapezoid intersect at a point that divide them into two sections that are proportional to the quotients of the bases.

4.- The sum of the squares of the diagonals of a trapezoid is equal to the sum of the squares of its sides plus the double product of its bases.

5.- The segment that joins the midpoints of the diagonals has a length equal to the half-difference of the bases.

6.- The angles adjacent to the lateral ones are supplementary.

7.- In a scalene trapezoid, the lengths of its diagonals are different.

8.- A trapezoid has an inscribed circumference only if the sum of its bases is equal to the sum of its sides.

9.- If a trapezoid has an inscribed circumference, then the angle with the vertex in the center of said circumference and sides that pass through the ends of the side of the trapezoid is straight.

10.- A scalene trapezoid does not have a circumscribed circumference, the only type of trapezoid that does is isosceles.

Formulas and equations

The following relationships of the scalene trapezoid are referred to the following figure.

1.- If AE = ED and BF = FC → EF - AB and EF - DC.

2.- EF = (AB + DC) / 2 that is: m = (a + c) / 2.

3. DI = IB = d ₁ /2 and AG = GC = d ₂ /2.

4.- DJ / JB = (c / a) similarly CJ / JA = (c / a).

Figure 3. Median and diagonals of a scalene trapezoid. Source: F. Zapata.

5.- DB ² + AC ² = AD ² + BC ² + 2 AB ∙ DC

Equivalently:

d ₁ ² + d ₂ ² = d ² + b ² + 2 a ∙ c

6.- GI = (AB - DC) / 2

That is to say:

n = (a - c) / 2

7.- α + δ = 180⁰ and β + γ = 180⁰

8.- If α ≠ β ≠ γ ≠ δ then d1 ≠ d2.

9.- Figure 4 shows a scalene trapezoid that has an inscribed circumference, in this case it is true that:

a + c = d + b

10.- In a scalene trapezoid ABCD with an inscribed circumference of center O, the following is also true:

∡AOD = ∡BOC = 90⁰

Figure 4. If in a trapezoid it is verified that the sum of its bases is equal to the sum of the sides, then there is a circumference inscribed in it. Source: F. Zapata.

Height

The height of a trapezoid is defined as the segment that goes from a point of the base perpendicularly to the opposite base (or its extension).

All the heights of the trapezoid have the same measurement h, so most of the time the word height refers to its measurement. In short, height is the distance or separation between the bases.

The height h can be determined by knowing the length of one side and one of the angles adjacent to the side:

h = d Sen (α) = d Sen (γ) = b Sen (β) = b Sen (δ)

Median

The measure m of the median of the trapezoid is the semi-sum of the bases:

m = (a + b) / 2

Diagonals

d ₁ = √

d ₂ = √

It can also be calculated if only the length of the sides of the trapezoid is known:

d ₁ = √

d ₂ = √

Perimeter

The perimeter is the total length of the contour, that is, the sum of all its sides:

P = a + b + c + d

Area

The area of ​​a trapezoid is the semisum of its bases multiplied by its height:

A = h ∙ (a + b) / 2

It can also be calculated if the median m is known and the height h:

A = m ∙ h

In case only the length of the sides of the trapezoid is known, the area can be determined using Heron's formula for the trapezoid:

A = ∙ √

Where s is the semiperimeter: s = (a + b + c + d) / 2.

Other ratios for the scalene trapezium

The intersection of the median with the diagonals and the parallel that passes through the intersection of the diagonals gives rise to other relationships.

Figure 5. Other relationships for the scalene trapezium. Source: F. Zapata.

-Relationships for the median EF

EF = (a + c) / 2; EG = IF = c / 2; EI = GF = a / 2

-Relationships for the segment parallel to the bases KL, and passing through the intersection point J of the diagonals

If KL - AB - DC with J ∈ KL, then KJ = JL = (a ∙ c) / (a ​​+ c)

Construction of the scalene trapezoid with ruler and compass

Given the bases of lengths a and c, where a> cy with sides of lengths b and d, where b> d, proceed by following these steps (see figure 6):

1.- With the rule the segment of the major AB is drawn.

2.- From A se and on AB mark point P so that AP = c.

3.- With the compass with center in P and radius d an arc is drawn.

4.- A center is made at B with radius b by drawing an arc that intersects the arc drawn in the previous step. We call Q the point of intersection.

Figure 6. Construction of a scalene trapezoid given its sides. Source: F. Zapata.

5.- With the center at A, draw an arc of radius d.

6.- With the center at Q, draw an arc of radius c that intercepts the arc drawn in the previous step. The cut-off point will be called R.

7.- Segments BQ, QR and RA are drawn with the ruler.

8.- Quadrilateral ABQR is a scalene trapezoid, since APQR is a parallelogram, which guarantees that AB - QR.

Example

The following lengths are given in cm: 7, 3, 4 and 6.

a) Determine if with them it is possible to construct a scalene trapezoid that can circumscribe a circle.

b) Find the perimeter, the area, the length of the diagonals and the height of said trapezoid, as well as the radius of the inscribed circle.

- Solution to

Using the segments of length 7 and 3 as bases and those of length 4 and 6 as sides, a scalene trapezoid can be constructed using the procedure described in the previous section.

It remains to check if it has an inscribed circumference, but remembering the property (9):

We see that effectively:

7 + 3 = 4 + 6 = 10

Then the condition of existence of inscribed circumference is satisfied.

- Solution b

Perimeter

The perimeter P is obtained by adding the sides. Since the bases add up to 10 and the lateral ones also, the perimeter is:

P = 20 cm

Area

To determine the area, known only its sides, the relationship is applied:

A = ∙ √

Where s is the semiperimeter:

s = (a + b + c + d) / 2.

In our case, the semiperimeter is worth s = 10 cm. After substituting the respective values:

a = 7 cm; b = 6 cm; c = 3 cm; d = 4 cm

Remains:

A = √ = (5/2) √63 = 19.84 cm².

Height

The height h is related to the area A by the following expression:

A = (a + c) ∙ h / 2, from which the height can be obtained by clearing:

h = 2A / (a ​​+ c) = 2 * 19.84 / 10 = 3.988 cm.

Radius of the inscribed circle

The radius of the inscribed circle is equal to half the height:

r = h / 2 = 1,984 cm

Diagonals

Finally we find the length of the diagonals:

d ₁ = √

d ₂ = √

Properly substituting the values ​​we have:

d ₁ = √ = √ (36 + 21-7 (20) / 4) = √ (22)

d ₂ = √ = √ (16 + 21-7 (-20) / 4) = √ (72)

That is: d ₁ = 4.69 cm and d ₂ = 8.49 cm

Figure 7. Scalene trapezoid that meets the condition of existence of an inscribed circumference. Source: F. Zapata.

Exercise resolved

Determine the interior angles of the trapezoid with bases AB = a = 7, CD = c = 3 and lateral angles BC = b = 6, DA = d = 4.

Solution

The cosine theorem can be applied to determine the angles. For example, the angle ∠A = α is determined from the triangle ABD with AB = a = 7, BD = d2 = 8.49, and DA = d = 4.

The cosine theorem applied to this triangle looks like this:

d ₂ ² = a ² + d ² - 2 ∙ a ∙ d ∙ Cos (α), that is:

72 = 49 + 16-56 ∙ Cos (α).

Solving for, the cosine of angle α is obtained:

Cos (α) = -1/8

That is, α = ArcCos (-1/8) = 97.18⁰.

The other angles are obtained in the same way, their values ​​being:

β = 41.41⁰; γ = 138.59⁰ and finally δ = 82.82⁰.

References

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