Question

The diagonal of a rectangle measures 8√2 inches. If the width is 8inches less than the length, then find the dimensions of the rectangle

Answer 1

We have the following situation:

• The diagonal of a rectangle is equal to 8√2 inches.

,

• The width is 8 inches less than the length

And we need to find the dimensions of the rectangle.

Then we can proceed as follows:

1. We know that all the internal angles of a rectangle are right angles, and the diagonals are congruent. We also know that the width of this rectangle is 8 inches less than the length:

`\begin{gathered} w=l-8 \\ \\ d=8√(2) \end{gathered}`

2. Then we can draw the situation as follows:

3. Now, we can apply the Pythagorean Theorem as follows:

`\begin{gathered} (8√(2))^2=l^2+w^2=l^2+(l-8)^2 \\ \\ \text{ Then we have:} \\ \\ l^2+(l-8)^2=(8√(2))^2 \\ \\ l^2+(l-8)^2=8^2(2) \\ \\ l^2+(l-8)^2=64(2)=128 \\ \\ l^2+(l-8)^2=128 \end{gathered}`

4. We have to expand the binomial expression on the left side of the equation:

`\begin{gathered} l^2+l^2-2(8)(l)+8^2=128 \\ \\ 2l^2-16l+64=128 \\ \\ 2l^2-16t+64-128=0 \\ \\ 2l^2-16l-64=0 \end{gathered}`

5. Now, we need to apply the quadratic formula to find the value of l as follows:

`x=(-b\pm√(b^2-4ac))/(2a),ax^2+bx+c=0`

6. Then we have that:

`\begin{gathered} 2l^(2)-16l-64=0 \\ \\ a=2,b=-16,c=-64 \\ \\ \text{ Then we have:} \\ \\ x=(-(-16)\pm√((-16)^2-4(2)(-64)))/(2(2)) \\ \\ x=(16\pm√(256+512))/(2(2))=(16\pm√(768))/(4) \end{gathered}`

7. Now, to simplify the radicand, we need to find the factors of 768:

`768=2^8*3`

Now, we have:

`\begin{gathered} √(768)=√(2^8*3)=(2^8*3)^{(1)/(2)}=2^{(8)/(2)}*3^{(1)/(2)}=2^4*√(3)=16√(3) \\ \\ √(768)=16√(3) \end{gathered}`

8. Then the values for l are two possible ones:

`\begin{gathered} x=(16\pm√(768))/(4)=(16\pm16√(3))/(4) \\ \\ x=(16+16√(3))/(4),x=(16-16√(3))/(4) \\ \\ x_1=(16)/(4)(1+√(3)),x_2=(16)/(4)(1-√(3)) \\ \\ x_1=4(1+√(3)),x_2=4(1-√(3)) \end{gathered}`

We can see that x2 gives us a negative value, and since we are finding a length, which is a positive value, then the value for l is:

`\begin{gathered} x_1=4(1-√(3))\approx−2.92820323028 \\ \\ x_2=4(1+√(3))\approx10.9282032303\text{ }\rightarrow\text{ This is the value we are finding.} \\ \text{ This is positive.} \end{gathered}`

9. Now, we have that:

`\begin{gathered} l=4(1+√(3)) \\ \\ \text{ Since }w=l-8,\text{ then we have:} \\ \\ w=4(1+√(3))-8=4+4√(3)-8=4-8+4√(3)=-4+4√(3) \\ \\ w=4(-1+√(3))=4(√(3)-1)\approx2.92820323028 \\ \\ w=4(√(3)-1) \end{gathered}`

10. Now, we can check both values as follows:

`\begin{gathered} l^2+w^2=d^2 \\ \\ (4(1+√(3))^)^2+(4(√(3)-1))^2=(8√(2))^2 \\ \\ 128=128\text{ }\rightarrow\text{This result is always true.} \end{gathered}`

Therefore, in summary, the dimensions of the rectangle are:

`\begin{gathered} l=4(1+√(3))=4(√(3)+1) \\ \\ w=4(√(3)-1) \end{gathered}`