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Triangles

Short Answer Type

421. In triangle ABC, D and E are points on side AB such that AD = BE. If DP || BC and EQ || AC, prove that PQ || AB.

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422.

In the given fig, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ BAC = 90°, AD ⊥ BC. Prove that AB² + CD² = BD² + AC².

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Long Answer Type

423. Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In the Fig. ABCD is a rhombus. Prove that 4AB² = AC² + BD².

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424. Any point X insides ∆DEF is joined to its vertices from a point P in DX, PQ is drawn parallel to EF meeting XF at R. Prove that PR || DF.

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425. In a right ∆ABC, the perpendicular BD on hypotenuse AC is drawn. Prove that AC.CD = BC².

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Short Answer Type

426. In ∆ABC, AD is a median and E, is mid-point of AD. If BE is produced, it meets AC at F. Show that AF =

1 third AC.

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427. P is a point in the interior of rectangle ABCD. If P is joined of each of the vertices of the rectangle. Prove that PB² + PD² = PA^{2 ;}PC².

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428. In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3. Prove that 9AD² = 7AB².

Solution not provided.

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Long Answer Type

429. In a right-angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides prove it using the above prove the following:
In ∆ABC, D is the mid-point of BC and AE ⊥ BC. If AC > AB, show that AB² = AD²- BC