Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362592 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0722 1212032 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0722 1212032 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1456 4049944 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1471 4077624 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1478 4095336 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1480 4126368 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1491 4149088 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1491 4149088 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2395 4773736 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2423 4874720 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2430 4895648 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2540 4962040 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2541 4962296 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Trapzium Cut in Half By Line Parallel to the Parallel Sides

Suppose we have a trapezium with parallel sides 10 and 4, and height 4. The trapezium is to be cut in half by a line parallel to the parallel sides. Let the distance of this line from the longer parallel side be  
\[x\]
.

We must find an equation in terms of  
\[x\]
  for the length  
\[l\]
  of the cutting line. When  
\[x=0, \: l=10\]
  and when  
\[x=4, \: l=4\]
.
From these two,  
\[l=10-1.5x\]

The area of the part above the cutting line is  
\[\frac{1}{2}(4+10-1.5x)(4-x)\]

The area of the part below the cutting line is  
\[\frac{1}{2}(10+10-1.5x)x\]

Equating these gives
\[\frac{1}{2}(4+10-1.5x)(4-x)=\frac{1}{2}(10+10-1.5x)x\]

Expanding this and simplifying gives  
\[3x^2-40x+56=0\]

Solving this equation gives  
\[x= \frac{40 \pm \sqrt{928}}{6}\]

Obviously x must be less than the height of the trapezium so  
\[x= \frac{40 - \sqrt{928}}{6}= x= \frac{20 - 2\sqrt{58}}{3}\]
  since the other possibility gives  
\[x \gt 4\]
.