Respuesta :

gmany
Look at the picture.
Use the Pythagorean theorem:
[tex]|AD|^2=5^2+(\sqrt{75})^2\\\\|AD|^2=25+75\\\\|AD|^2=100\to|AD|=\sqrt{100}\to|AD|=10[/tex]
The area of a triangle ADC:
[tex]A_\Delta=\dfrac{|DC|\cdot|AC|}{2}\ and\ A_\Delta=\dfrac{|AD|\cdot h}{2}[/tex]
Substitute:
[tex]\dfrac{10h}{2}=\dfrac{5\sqrt{75}}{2}\\\\5h=\dfrac{5\sqrt{75}}{2}\ \ \ |:5\\\\h=\dfrac{\sqrt{75}}{2}[/tex]
The triangles ADC and DCE are similar. Therefore:
[tex]\dfrac{|ED|}{\frac{\sqrt{75}}{2}}=\dfrac{5}{\sqrt{75}}\\\\\dfrac{2|ED|}{\sqrt{75}}=\dfrac{5}{\sqrt{75}}\ \ \ |\cdot\sqrt{75}\\\\2|ED|=5\ \ \ |:2\\\\|ED|=2.5[/tex]
[tex]|BC|=|AD|-2|ED|\\\\|BC|=10-2\cdot2.5=10-5=5[/tex]
The area of the trapezoid:
[tex]A=\dfrac{|AD|+|BC|}{2}\cdot h[/tex]
Substitute:
[tex]A=\dfrac{10+5}{2}\cdot\dfrac{\sqrt{75}}{2}=\dfrac{15\sqrt{75}}{4}\\\\=\dfrac{15\sqrt{25\cdot3}}{4}=\dfrac{15\cdot5\sqrt3}{4}=\dfrac{75\sqrt3}{4}[/tex]

Ver imagen gmany

The area of a 2D form is the amount of space within its perimeter. The area of the given trapezoid ABCD is (75√3)/2 sq. units.

What is an area?

The area of a 2D form is the amount of space within its perimeter. It is measured in square units such as cm2, m2, and so on. To find the area of a square formula or another quadrilateral, multiply its length by its width.

In the given trapezoid ABCD,  For the right-angled ΔACD, the value of AD can be written as,

AD² = AC² + CD²

AD² = (√75)² + 5²

AD² = 100

AD = 10 units

Also, In the given trapezoid ABCD,  the area of ΔACD can be found in two ways,

Area ΔACD = (0.5×AC×CD) = (0.5×CE×AD)

Solving the above equation for CE,

(0.5×AC×CD) = (0.5×CE×AD)

AC×CD = CE×AD

√75 × 5 = CE × 10

CE = (√75)/2 units

Now, in ΔCDE, using the Pythagoras theorem, we can write,

CD² = CE² + ED²

5² = (√75/2)² + ED²

25-(75/4) = ED²

ED² = 6.25

ED = 2.5 units

Now, the length of side BC can be written as,

BC = AD - 2(ED)

BC = 10 - 2(2.5)

BC = 5 units

Now, the area of the trapezoid ABCD can be written as,

Area = 0.5 × (AD+BC ) × CE

Area = 0.5 × (10+5) × (√75/2)

Area = (15√75)/2

Area = (75√3)/2 sq. units

Hence, the area of the given trapezoid ABCD is (75√3)/2 sq. units.

Learn more about Area:

https://brainly.com/question/1631786

#SPJ2

Ver imagen ap8997154

Otras preguntas