What is the area of a figure with vertices(1,1), (8,1), and (5,5)?

It will be a triangle

Calculating sides of this triangle

A(1,1)

B(8,1)

C(5,5)

side a-AB=\( \sqrt{(8-1)^{2}+( 1-1)^{2} ) }= \sqrt{ 7^{2} }=7 \)

side b-AC=\( \sqrt{(5-1)^{2}+( 5-1)^{2} ) }= \sqrt{ 4^{2}+4^{2} }= \sqrt{36} \)=6 )

side c-BC=\( \sqrt{(5-8)^{2}+( 5-1)^{2} ) }= \sqrt{ (-3)^{2}+4^{2} }= \sqrt{25} \)=5 )

Area of the triangle from Herone formula:

A=\( \sqrt{p(p-a)(p-b)(p-c)} \)

p=\( \frac{1}{2}(a+b+c) \)=\( \frac{1}{2}(7+6+5) \)=9

A=\( \sqrt{9(9-7)(9-6)(9-5)} \)=\( \sqrt{9*2*3*4} \)=\( \sqrt{216} \)

Area is \( \sqrt{216} \)

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