AMC 10 Math Olympiad USA Math Olympiad

Area of Square – Singapore Mathematical Olympiad – 2013 – Problem No.17

Try this beautiful problem from Singapore Mathematical Olympiad. 2013 based on the area of Square.

Problem – Area of Square

Let ABCD be a square and X and Y be points such that the lengths of XY, AX, and AY are 6,8 and 10 respectively. The area of ABCD can be expressed as \(\frac{m}{n}\) units where m and n are positive integers without common factors. Find the value of m+n.

area of square
  • 1215
  • 1041
  • 2001
  • 1001

Key Concepts

2D Geometry

Area of Square

Check the Answer

Answer: 1041

Singapore Mathematical Olympiad – 2013 – Junior Section – Problem Number 17

Challenges and Thrills –

Try with Hints

This can the very first hint to start this sum:

Assume the length of the side is a.

Now from the given data we can apply Pythagoras’ Theorem :

Since, \(6^2+8^2 = 10^2\)

so \(\angle AXY = 90^\circ\).

From this, we can understand that \(\triangle ABX \) is similar to \(\triangle XCY\)

Try to do the rest of the sum……………………

From the previous hint we find that :

\(\triangle ABX \sim \triangle XCY\)

From this we can find \(\frac {AX}{XY} = \frac {AB}{XC} \)

\(\frac {8}{6} = \frac {a}{a – BX}\)

Can you now solve this equation ?????????????

This is the very last part of this sum :

Solving the equation from last hint we get :

a = 4BX and from this we can compute :

\(8^2 = {AB}^2 +{BX}^2 = {16BX}^2 + {BX}^2 \)

so , \( BX = \frac {8}{\sqrt {17}} and \(a^2 = 16 \times \frac {64}{17} = \frac {1024}{17}\)

Thus m + n = 1041 (Answer).

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AMC 10

Problem related to triangle – AMC 10B, 2019 Problem 10

The given problem is related to the calculation of area of triangle and distance between two points.

Try the problem

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?

$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

2019 AMC 10B Problem 10

Problem related to triangle

6 out of 10

Secrets in Inequalities.

Knowledge Graph

Problem related to triangle- knowledge graph

Use some hints

Notice that it does not matter where the triangle is in the 2D plane so for our easy access we can select two points A and B in any place of choice.

So we can actually select any two points A and B such that they are 10 units apart so lets the points are \(A(0,0)\) and \(B(10,0)\) , as they are 10 units apart.

Now we can select the point C such that the perimeter of the triangle is 50 units. and then we can apply the formula of area to calculate the possible positions of C.

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