10th Class Mathematics Solved Paper - Mathematics-2016 Outside Delhi Set-I

  • question_answer Prove that the points (3, 0), (6, 4) and (\[1,\text{ }3\]) are the vertices of a right angled isosceles triangle.

    Answer:

    Let \[A(3,0),\text{ }B(6,4)\] and \[C(-1,3)\] be the vertices of a triangle \[ABC\].
    Length of                       \[AB=\sqrt{{{(6-3)}^{2}}+{{(4-0)}^{2}}}\]
                                        \[=\sqrt{{{(3)}^{2}}+{{(4)}^{2}}}\]
                                        \[=\sqrt{9+16}=\sqrt{25}=5\,\]units
    Length of                       \[BC=\sqrt{{{(-1-6)}^{2}}+{{(3-4)}^{2}}}\]
                                        \[=\sqrt{{{(-7)}^{2}}+{{(-1)}^{2}}}\]
                                        \[=\sqrt{49+1}=\sqrt{50}=5\sqrt{2}\] units.
    And Length of    \[AC=\sqrt{{{(-1-3)}^{2}}+{{(3-0)}^{2}}}\]
                                        \[=\sqrt{{{(-4)}^{2}}+{{(3)}^{2}}}\]
                                        \[=\sqrt{16+9}=\sqrt{25}=5\] units
    \[\therefore \]                         \[AB=AC\]
    And         \[{{(AB)}^{2}}+{{(AC)}^{2}}={{(BC)}^{2}}\]
    Hence, \[\Delta \text{ }ABC\] is a isosceles, right angled triangle.                              Hence Proved


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