Find the area of a quadrilateral ABCD in which AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diagonal BD = 20 cm.
Understand the Problem
The question involves solving various problems related to the area of different geometric shapes, specifically quadrilaterals, trapeziums, and parallelograms, using given dimensions and properties.
Answer
The area of quadrilateral \( ABCD \) is \( 546 \, \text{cm}^2 \).
Answer for screen readers
The area of quadrilateral ( ABCD ) is ( 546 , \text{cm}^2 ).
Steps to Solve
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Identify the quadrilateral and properties In this problem, you are dealing with quadrilateral ( ABCD ) where ( AB = 42 , \text{cm} ), ( BC = 21 , \text{cm} ), ( CD = 29 , \text{cm} ), ( DA = 34 , \text{cm} ), and the diagonal ( BD = 20 , \text{cm} ).
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Use the formula for area For a quadrilateral given the lengths of its sides and one diagonal, you can use Brahmagupta's formula. The area ( A ) of quadrilateral ( ABCD ) can be calculated in two triangles ( ABD ) and ( BCD ):
$$ A = \frac{1}{2} \times d \times (h_1 + h_2) $$ where ( d ) is the diagonal and ( h_1, h_2 ) are the heights of triangles ( ABD ) and ( BCD ).
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Calculate the area of triangles using Heron’s Formula You will need to calculate the semi-perimeter ( s ) for each triangle:
For ( \triangle ABD ):
- Calculate the semi-perimeter: $$ s_1 = \frac{AB + AD + BD}{2} = \frac{42 + 34 + 20}{2} = 48 $$
- Calculate the area using Heron’s formula: $$ A_1 = \sqrt{s_1(s_1 - AB)(s_1 - AD)(s_1 - BD)} = \sqrt{48(48 - 42)(48 - 34)(48 - 20)} $$
For ( \triangle BCD ):
- Calculate the semi-perimeter: $$ s_2 = \frac{BC + CD + BD}{2} = \frac{21 + 29 + 20}{2} = 35 $$
- Calculate the area using Heron’s formula: $$ A_2 = \sqrt{s_2(s_2 - BC)(s_2 - CD)(s_2 - BD)} = \sqrt{35(35 - 21)(35 - 29)(35 - 20)} $$
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Sum the areas of the two triangles Finally, add the area of the two triangles to get the total area of quadrilateral ( ABCD ):
$$ A_{total} = A_1 + A_2 $$
The area of quadrilateral ( ABCD ) is ( 546 , \text{cm}^2 ).
More Information
This problem demonstrates the application of Heron's formula to find the area of triangles and then adds them up to find the area of a complex geometric shape like a quadrilateral.
Tips
- Not calculating the semi-perimeter correctly: Ensure to double-check that you are using the correct lengths for the semi-perimeter.
- Misapplying Heron’s Formula: Be cautious while substituting values into Heron's formula to avoid calculation errors.