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AMC 10 USA Math Olympiad

Area of Triangle – AMC 10A – 2019 – Problem No. – 7

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

What is Area of Triangle ?


The area of a triangle is defined as the total space that is enclosed by any particular triangle. The basic formula to find the area of a given triangle is A = 1/2 × b × h,  where b is the base and h is the height of the given triangle, whether it is scalene, isosceles or equilateral.

Try This Problem from AMC 10A – 2019 -Problem No.7


Two lines with slopes \(\frac{1}{2}\) and 2 intersect at (2,2) . What is the area of the triangle enclosed by these two lines and the line \(x + y = 10 \) ?

A) 4 B) \(4\sqrt 2\) C) 6 D) 8 E) \(6 \sqrt 2\)


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Suggested Book

American Mathematics Competition 10 (AMC 10A), 2019, Problem Number – 7

Area of Triangle

6 out of 10

Problems in Plane Geometry by Sharygin

Knowledge Graph


area of triangle - Knowledge Graph

Use some hints


First Hint

If you need a hint to start this sum use this


Lets try to find the slop – intercept form of all three lines : (x,y) = (2,2) and y =

\(\frac{x}{2}+b\) implies \(2 = \frac{2}{2}+b = 1+b\). So, b = 1 . While y = 2x + c implies 2 = 2.2 + c So, c = -2 And again x+y = 10 implies y = -x + 10.

Second Hint

Thus the lines are \( y = \frac {x}{2} + 1 \) , y = 2x – 2 and y = -x + 10 . Now we find the intersection points between each of the lines with y = -x + 10 , which are (6,4) and (4,6) .

Final Hint

In the last hint we can apply the distance formula and then the Pythagorean Theorem, we see that we have an isosceles triangle where the base is \(2\sqrt 2\) and the height \(3 \sqrt 2\), whose area is 6 .The answer is 6 (c) .

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