Rectangle ABCD: Express (X-Y)² Using Side Lengths and Triangle Area S

Question

Given the rectangle ABCD

AB=Y AD=X

The triangular area DEC equals S:

Express the square of the difference of the sides of the rectangle

using X, Y and S:

YYYXXXAAABBBCCCDDDEEE

Video Solution

Solution Steps

00:00 Express the square of rectangle side differences
00:11 Let's substitute side expressions according to the given data and open parentheses
00:20 Triangle area expression according to the given data
00:27 Triangle height equals rectangle side X
00:30 We'll use the formula for calculating triangle area
00:33 (height multiplied by base) divided by 2
00:40 Let's find the expression for X
00:46 Let's find the expression for Y
00:54 Let's find the expression for the product of sides
01:02 Let's substitute these expressions in our shortened multiplication formula
01:17 Pay attention to squaring both numerator and denominator
01:29 Let's take out the common factor from the parentheses
01:44 And this is the solution to the problem

Step-by-Step Solution

Since we are given the length and width, we will substitute them according to the formula:

(AD−AB)2=(x−y)2 (AD-AB)^2=(x-y)^2

x2−2xy+y2 x^2-2xy+y^2

The height is equal to side AD, meaning both are equal to X

Let's calculate the area of triangle DEC:

S=xy2 S=\frac{xy}{2}

x=2Sy x=\frac{2S}{y}

y=2Sx y=\frac{2S}{x}

xy=2S xy=2S

Let's substitute the given data into the formula above:

(2Sy)2−2×2S+(25x)2 (\frac{2S}{y})^2-2\times2S+(\frac{25}{x})^2

4S2y2−4S+4S2x2 \frac{4S^2}{y^2}-4S+\frac{4S^2}{x^2}

4S=(S2y+S2x−1) 4S=(\frac{S^2}{y}+\frac{S^2}{x}-1)

Answer

(x−y)2=4s[sy2+sx2−1] (x-y)^2=4s\lbrack\frac{s}{y^2}+\frac{s}{x^2}-1\rbrack

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Related Subjects

The Area of a Triangle Calculating the Area of a Rectangle The formula for the difference of squares Area Area of Equilateral Triangles Area of a Scalene Triangle Area of Isosceles Triangles Rectangle Abbreviated Multiplication Formulas Triangle Area of a right triangle

Question Types

Square of Difference: Using quadrilaterals Area of a Rectangle: Using short multiplication formulas