Question

the perimeter of an isosceles trapezoid is 62 cm. if three sides are equal in length and the fourth side is 10 cm longer, find the area of the trapezoid.

Answer 1

Perimeter = sum of all the sides
let x = one side
longer side = x + 10
Perimeter = x + x + x + x + 10
62 = 4x + 10
4x = 52
x = 13
Longer side = 23

Area = (10+23)x2/10
         = 6.6  

Answer 2

The area of trapezoid is $\boxed{216{\text{ c}}{{\text{m}}^2}}.$

Further explanation:

The formula of area of trapezoid can be expressed as follows,

$\boxed{{\text{Area of trapezoid}} = \frac{1}{2} \times \left( {{\text{sum of parallel sides}}} \right) \times {\text{height}}}$

Given:

The perimeter of an isosceles trapezoid is 62 cm.

The length of the fourth side is $10{\text{ cm}}.$

Explanation:

Consider the length of the side of trapezoid as $x.$

Consider the longer side of the trapezoid as $x+10.$

Perimeter of trapezoid is 62 cm.

$\begin{aligned}{\text{Perimeter}}&= 62{\text{ cm}}\\x + x + x + x + 10 &= 62\\4x + 10 &= 62\\4x &= 62 - 10\\4x&= 52\\x&= \frac{{52}}{4}\\x&= 13\\\end{aligned}$

The longer side of the trapezoid is $23{\text{ cm}}.$

The semi perimeter of triangle ABE can be obtained as follows,

$\begin{aligned}S&= \frac{{a + b + c}}{2}\\&= \frac{{13 + 13 + 10}}{2}\\&= 18\\\end{aligned}$

The area of triangle ABE can be obtained as follows,

$\begin{aligned}{\text{Area of }}\Delta ABE &= \sqrt {S \times \left( {S - a} \right) \times \left( {S - b} \right) \times \left( {S - c} \right)}\\&= \sqrt {18 \times \left( {18 - 13} \right)\left( {18 - 13} \right)\left( {18 - 10} \right)} \\&= \sqrt {18 \times 5 \times 5 \times 8}\\&= 5 \times12\\&= 60\\\end{aligned}$

The height of the trapezoid can be obtained as follows,

$\begin{aligned}{\text{Area} &= 60\\\frac{1}{2}\times b\times h&= 60\\10 \timesh&= 120\\h&= 12\\\end{aligned}$

The height of the trapezoid is 12 cm.

The area of parallelogram ABED can be obtained as follows,

$\begin{aligned}{\text{Area of ABED}}&= b\times h\\& = 13 \times12\\&= 156{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

The area of trapezoid can be calculated as follows,

$\begin{aligned}{\text{Area of ABCD}}&= {\text{Area of ABED}} + {\text{Area of BCE}} \\{\text{}}&=156 + 60\\&= 216{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

The area of trapezoid is $\boxed{216{\text{ c}}{{\text{m}}^2}}.$

Kindly refer to the image attached.

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Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Triangles

Keywords: triangle, triangle pair, equal angles, sides, area, trapezoid, two triangles, bases, intersecting, diagonal, segment, sector, minor segment.