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## 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.

• 1215
• 1041
• 2001
• 1001

### Key Concepts

2D Geometry

Area of Square

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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## 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.

## 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.