IDNLearn.com provides a collaborative environment for finding and sharing answers. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.
Sagot :
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:
[tex]\begin{gathered} w=l-8 \\ \\ d=8\sqrt{2} \end{gathered}[/tex]2. Then we can draw the situation as follows:
3. Now, we can apply the Pythagorean Theorem as follows:
[tex]\begin{gathered} (8\sqrt{2})^2=l^2+w^2=l^2+(l-8)^2 \\ \\ \text{ Then we have:} \\ \\ l^2+(l-8)^2=(8\sqrt{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}[/tex]4. We have to expand the binomial expression on the left side of the equation:
[tex]\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}[/tex]5. Now, we need to apply the quadratic formula to find the value of l as follows:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},ax^2+bx+c=0[/tex]6. Then we have that:
[tex]\begin{gathered} 2l^{2}-16l-64=0 \\ \\ a=2,b=-16,c=-64 \\ \\ \text{ Then we have:} \\ \\ x=\frac{-(-16)\pm\sqrt{(-16)^2-4(2)(-64)}}{2(2)} \\ \\ x=\frac{16\pm\sqrt{256+512}}{2(2)}=\frac{16\pm\sqrt{768}}{4} \end{gathered}[/tex]7. Now, to simplify the radicand, we need to find the factors of 768:
[tex]768=2^8*3[/tex]Now, we have:
[tex]\begin{gathered} \sqrt{768}=\sqrt{2^8*3}=(2^8*3)^{\frac{1}{2}}=2^{\frac{8}{2}}*3^{\frac{1}{2}}=2^4*\sqrt{3}=16\sqrt{3} \\ \\ \sqrt{768}=16\sqrt{3} \end{gathered}[/tex]8. Then the values for l are two possible ones:
[tex]\begin{gathered} x=\frac{16\pm\sqrt{768}}{4}=\frac{16\pm16\sqrt{3}}{4} \\ \\ x=\frac{16+16\sqrt{3}}{4},x=\frac{16-16\sqrt{3}}{4} \\ \\ x_1=\frac{16}{4}(1+\sqrt{3}),x_2=\frac{16}{4}(1-\sqrt{3}) \\ \\ x_1=4(1+\sqrt{3}),x_2=4(1-\sqrt{3}) \end{gathered}[/tex]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:
[tex]\begin{gathered} x_1=4(1-\sqrt{3})\approx−2.92820323028 \\ \\ x_2=4(1+\sqrt{3})\approx10.9282032303\text{ }\rightarrow\text{ This is the value we are finding.} \\ \text{ This is positive.} \end{gathered}[/tex]9. Now, we have that:
[tex]\begin{gathered} l=4(1+\sqrt{3}) \\ \\ \text{ Since }w=l-8,\text{ then we have:} \\ \\ w=4(1+\sqrt{3)}-8=4+4\sqrt{3}-8=4-8+4\sqrt{3}=-4+4\sqrt{3} \\ \\ w=4(-1+\sqrt{3})=4(\sqrt{3}-1)\approx2.92820323028 \\ \\ w=4(\sqrt{3}-1) \end{gathered}[/tex]10. Now, we can check both values as follows:
[tex]\begin{gathered} l^2+w^2=d^2 \\ \\ (4(1+\sqrt{3})^)^2+(4(\sqrt{3}-1))^2=(8\sqrt{2})^2 \\ \\ 128=128\text{ }\rightarrow\text{This result is always true.} \end{gathered}[/tex]Therefore, in summary, the dimensions of the rectangle are:
[tex]\begin{gathered} l=4(1+\sqrt{3})=4(\sqrt{3}+1) \\ \\ w=4(\sqrt{3}-1) \end{gathered}[/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.
