Respuesta :
Answer:
The correct option is 1.
Step-by-step explanation:
Given information: The coordinates of a right angled triangle ABC are A(0, 0), B(0, 4a – 5) and C(2a + 1, 2a + 6). Angle ABC = 90°.
It means AB and BC are legs of the right angled triangle ABC.
Side AB lies on the y-axis because the x-coordinate of both A and B is 0.
Two legs are perpendicular to each other. So, BC must be parallel to x-axis and the y-coordinate of both B and C is must be same.
[tex]4a-5=2a+6[/tex]
[tex]4a-2a=5+6[/tex]
[tex]2a=11[/tex]
Divide both sides by 2.
[tex]a=\frac{11}{2}[/tex]
The value of a is 2. So the coordinates of triangle ABC are
[tex]B(0,4a-5)=B(0,4(\frac{11}{2})-5)\Rightarrow B(0,17)[/tex]
[tex]C(2a+1,2a+6)=C(2(\frac{11}{2})+1,2(\frac{11}{2})+6)\Rightarrow C(12,17)[/tex]
The area of a triangle is
[tex]Area=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]
The area of triangle ABC is
[tex]Area=\frac{1}{2}|0(17-17)+0(17-0)+12(0-17)|[/tex]
[tex]Area=\frac{1}{2}|12(-17)|[/tex]
[tex]Area=\frac{1}{2}|-204|[/tex]
[tex]Area=\frac{1}{2}(204)[/tex]
[tex]Area=102[/tex]
The area of triangle ABC is 102. Therefore the correct option is 1.
