Area of Triangle Problem | AMC-8, 2019

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Try this beautiful problem from Geometry based on the area of the triangle.

Area of Triangle – AMC-8, 2019- Problem 21

What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$

Geometry

Triangle

Linear equation

But try the problem first…

Answer: $16$

Source

First hint

Find the three vertex of the triangle

Can you now finish the problem ……….

Second Hint

The area of the Triangle =$\frac{1}{2} \times \{x_1(y_2 – y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}$

can you finish the problem……..

Final Step

Solving two The lines y=5 and y=1+x are intersect at (4,5)=$(x_1,y_1)$(say)

Solving two The lines y=5 and y=1-x are intersect at (-4,5)=$(x_2,y_2)$(say)

Solving two The lines y=1-x and y=1+x are intersect at (0,1)=$(x_1,y_1)$(say)

Then the area of Triangle =$\frac{1}{2} \times\{ x_1(y_2 – y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}$

= $\frac{1}{2} \times \{4(5-1)+(-4)(1-5)+0(5-5)=\frac{1}{2} (16+16)=16\}$