Examples with solutions for Product Property of Square Roots: Using additional geometric shapes

Exercise #1

Look at the following rectangle:

AAABBBCCCDDDEEE84

ΔAEB is isosceles (AE=EB).

Calculate the perimeter of the rectangle ABCD.

Video Solution

Step-by-Step Solution

Answer

8+163 8+16\sqrt3

Exercise #2

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal tox2 \sqrt{\frac{x}{2}}

We mark the length of the diagonal A A with m m

Check the correct argument:

XXXmmmAAABBBCCCDDD

Video Solution

Step-by-Step Solution

Let's find side BC

Based on what we're given:

ABBC=xBC=x2 \frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

2x=xBC \sqrt{2}x=\sqrt{x}BC

Let's divide by square root x:

2×xx=BC \frac{\sqrt{2}\times x}{\sqrt{x}}=BC

2×x×xx=BC \frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC

Let's reduce the numerator and denominator by square root x:

2x=BC \sqrt{2}\sqrt{x}=BC

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

AB2+BC2=AC2 AB^2+BC^2=AC^2

Let's substitute what we're given:

x2+(2x)2=m2 x^2+(\sqrt{2}\sqrt{x})^2=m^2

x2+2x=m2 x^2+2x=m^2

Answer

x2+2x=m2 x^2+2x=m^2

Exercise #3

Given the rectangle ABCD

AB=X

The ratio between AB and BC is x2 \sqrt{\frac{x}{2}}

We mark the length of the diagonal A the rectangle in m

Check the correct argument:

XXXmmmAAABBBCCCDDD

Video Solution

Step-by-Step Solution

Given that:

ABBC=x2 \frac{AB}{BC}=\sqrt{\frac{x}{2}}

Given that AB equals X

We will substitute accordingly in the formula:

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

x2=BCx x\sqrt{2}=BC\sqrt{x}

x2x=BC \frac{x\sqrt{2}}{\sqrt{x}}=BC

x×x×2x=BC \frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC

x×2=BC \sqrt{x}\times\sqrt{2}=BC

Now let's focus on triangle ABC and use the Pythagorean theorem:

AB2+BC2=AC2 AB^2+BC^2=AC^2

Let's substitute the known values:

x2+(x×2)2=m2 x^2+(\sqrt{x}\times\sqrt{2})^2=m^2

x2+x×2=m2 x^2+x\times2=m^2

We'll add 1 to both sides:

x2+2x+1=m2+1 x^2+2x+1=m^2+1

(x+1)2=m2+1 (x+1)^2=m^2+1

Answer

m2+1=(x+1)2 m^2+1=(x+1)^2