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AMC-8 Geometry Math Olympiad USA Math Olympiad

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

Try this beautiful problem from Geometry: The area of triangle AMC-8, 2019. You may use sequential hints to solve the problem

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 $y=1-x$

  • $8$
  • $ 16$
  • $15$

Key Concepts


Geometry

Triangle

Linear equation

Check the Answer


But try the problem first…

Answer: $16$

Source
Suggested Reading

AMC-8 (2019) Problem 21

Pre College Mathematics

Try with Hints


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

Area of the triangle

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 =\(\frac{1}{2} \times\{ x_1(y_2 – y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}\)

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

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