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Problem 3

# The sides of a triangle are $$3 x+4 y, 4 x+3 y$$ and $$5 x+5 y$$ units, where $$x>0, y>0$$. The triangle is: (a) right angled (b) acute angled (c) obtuse angled (d) isosceles

Expert verified
The triangle is a right angled triangle, but not an isosceles triangle.
See the step by step solution

## Step 1: Calculating the squares of the sides

First, calculate the square of each given side length. $(3x + 4y)^2 = 9x^2 + 24xy + 16y^2$$(4x + 3y)^2 = 16x^2 + 24xy + 9y^2$$(5x + 5y)^2 = 25x^2 + 50xy + 25y^2$

## Step 2: Comparing the sides

The next step is to compare the squares of the sides of the triangle to determine the type of triangle. From the calculations in step 1 we can see that \((5x + 5y)^2 = (3x + 4y)^2 + (4x + 3y)^2\], which means this triangle satisfies the Pythagorean theorem. Hence, it's a right angled triangle.

## Step 3: Checking for isosceles

Finally, we need to check if this is an isosceles triangle (two sides of equal length). After checking, we can see that no two sides of this triangle are equal. Therefore, it's not an isosceles triangle.

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