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Triangles

Long Answer Type

321. Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle.

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322. Points P, Q, R and S in that order are dividing a line segment joining A(2, 6) and B(7, 4) in five equal parts. Find the co-ordinates of P and R.

68 Views

323. BL and CM are median of a triangle ABC right angled at A. Prove that 4(BL² + CM²) = 5BC².

1368 Views

324.

In the given Fig, ABC is a right triangle, right angled at B. AD and CE are two medians drawn from A and C respectively. If AC = 5 cm and $AD equals fraction numerator 3 square root of 5 over denominator 2 end fraction cm comma$ find the length of CE.

1571 Views

Short Answer Type

325. In an isosceles triangle ABC, with AB = AC, BD is perpendicular from B to the side AC. Prove that BD² - CD² = 2CD.AD.

6267 Views

Long Answer Type

326. ABC is a triangle in which AB = AC and D is any point in BC. Prove that AB² - AD² = BD.CD.

3434 Views

327. In a right triangle ABC, right angled at C.P. and Q are the points on the sides CA and CB respectively, which divdes these sides in the ratio 2:1. Prove that
(i) 9AQ² 9AC² + 4BC²(ii) 9BP² 9BC² + 4AC²(iii) 9(AQ² + BP²) = 13AB².

1891 Views

328. In the given figure ABC is a triangle right angled at B. P and Q trisect BC. Prove that: 8AP² = 3AC² + 5AQ²

80 Views

329.
In the given Fig., ∠ACB = 90° and CD ⊥ AB. Prove that $BC squared over AC squared equals BD over AD.$

We know that,
If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
$rightwards double arrow space space space space space space space space space space increment ACD space tilde space increment ABC rightwards double arrow space space space space space space space space space space space space space space AC over AB space equals space AD over AC rightwards double arrow space space space space space space space space space space space space space space AC squared space equals space AD. space AB space space space space space space space space space... left parenthesis straight i right parenthesis$
And $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow space space space space space space space space space space space BC over BA equals BD over BC rightwards double arrow space space space space space space space space space space space space space BC squared space equals space BD space cross times space BA space space space space... left parenthesis ii right parenthesis$
From (i) and (ii)
$space BC squared over AC squared equals fraction numerator BD cross times BA over denominator AD cross times AB end fraction equals BD over AD$
Hence, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$