Square of sum: Using quadrilaterals

Question 1

The rectangle ABCD is shown below.

AB = X

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

The length of diagonal AC is labelled m.

Determine the value of m:

Question 2

Given the trapezoid where the height is equal to the sum of the two bases.

It is known that the difference between the large base and the small base is 5

We will mark the small base with X

Express the area of the trapezoid using X

Question 3

Given two squares, one side of the squares is larger by 2 than the other. The area of the large square is larger than the perimeter of the small square by 20

Find the length of the small square

Question 4

Given a rectangle whose side is greater by 6 than the other side. We mark the area of the rectangle with S

What is the correct argument?

Question 5

Given a square of side length X

We will mark the area of the square by S and the perimeter of the square by P

Check the correct statement

Examples with solutions for Square of sum: Using quadrilaterals

Exercise #1

The rectangle ABCD is shown below.

AB = X

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

The length of diagonal AC is labelled m.

Determine the value of m:

Video Solution

Step-by-Step Solution

We know that:

$\frac{AB}{BC}=\sqrt{\frac{x}{2}}$

We also know that AB equals X.

First, we will substitute the given data into the formula accordingly:

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

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

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

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

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

Now let's look at triangle ABC and use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

We substitute in our known values:

$x^2+(\sqrt{x}\times\sqrt{2})^2=m^2$

$x^2+x\times2=m^2$

Finally, we will add 1 to both sides:

$x^2+2x+1=m^2+1$

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

Answer

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

Exercise #2

Given the trapezoid where the height is equal to the sum of the two bases.

It is known that the difference between the large base and the small base is 5

We will mark the small base with X

Express the area of the trapezoid using X

Video Solution

Answer

$\frac{1}{2}\lbrack4x^2+20x+25\rbrack$

Exercise #3

Given two squares, one side of the squares is larger by 2 than the other. The area of the large square is larger than the perimeter of the small square by 20

Find the length of the small square

Video Solution

Answer

Exercise #4

Given a rectangle whose side is greater by 6 than the other side. We mark the area of the rectangle with S

What is the correct argument?

Video Solution

Answer

9+S equal to the smaller side plus 3 squared (the two squared).

Exercise #5

Given a square of side length X

We will mark the area of the square by S and the perimeter of the square by P

Check the correct statement

Video Solution

Answer

$P+S+4=(x+2)^2$

Question 1

Shown below is the rectangle ABCD.

AB = y

AD = x

Express the square of the sum of the sides of the rectangle using the area of the triangle DEC.

Exercise #6

Shown below is the rectangle ABCD.

AB = y

AD = x

Express the square of the sum of the sides of the rectangle using the area of the triangle DEC.

Video Solution

Answer

$(x+y)^2=4s\lbrack\frac{s}{y^2}+\frac{s}{x^2}+1\rbrack$