Answer to Question #319087 in Geometry for Nhlonipho Manganyi

Question

This question is concerning the completion of the square by using plane geometry

The area of a square of length x is : x

2

The area of a rectangle of sides x and y is : xy

Consider the expression : x

2 + 10x

Such can be considered as the sum of two areas viz square x

2 and a rectangle 10x

Now do the following (sketch essential)

Divide the rectangle into two equal areas viz 5x and 5x

Next to the square place the two rectangles with the side x in common. One to the RHS and the other below.

You will now have a ’nearly complete’ square

a) What is the length of each side of the ’missing square’ in your diagram

b) If you add the area of the missing square to make a big square, what is the area of the complete (big) square

c) By using the areas from your sketch, find the expressions ? and ?? in the following:

x

2 + 10x + (?)2 = (??)

Answer 1

Explanation:

The area of a rectangle can be calculated by multiplying the lengths of two adjacent sides. All of the choices given lists sufficient information, with one exception. We examine each of the choices.The lengths of one pair of adjacent sides: This choice is false, as is directly stated above.

The perimeter and the length of one side: Using the perimeter formula, you can find the length of an adjacent side, making this choice false.

The lengths of one side and a diagonal: using the Pythagorean Theorem, you can find the length of an adjacent side, making this choice false.

The lengths of one pair of opposite sides: this gives you no way of knowing the lengths of the adjacent sides. This is the correct choice.

A) The length of each side in will be (x²+10)×x²====> x⁴ + 10x²

B)you add the area of the missing square to make a big square, what is the area of the complete (big) square

4x² +10(4) +x² +10x

4x²+40+x²+10x===>5x²+10x+40

C) By using the areas from your sketch, find the expressions ? and ?? in the following:

x²+ 10x + (?)2 = (??)2 by comparing 5x²+10x+40 =0 it will be ? = 20 and

??= 0