Find angles in a right triangle if tan ⁡ θ = 3 4 , where θ is the angle between medians Find angles in right triangle if it is known that tan ⁡ θ = 3 4 , where θ is the angle between catheti medians.

Question

Find angles in a right triangle if   tan  ⁡  θ  =      3    4  , where   θ is the angle between medians Find angles in right triangle if it is known that   tan  ⁡  θ  =            3      4      , where   θ is the angle between catheti medians.

Quinn Hansen · Accepted Answer

Step 1In the picture, there are two edges that make the right angle. Call the horizontal one B and the vertical one A. The angle of the main triangle, that faces the edge A is called   α. The median partitions   α into two angles. The lower angle, I call it       α    1  .  t  a  n  (      α    1    )  =      A          2      B      Also, we have an angle, with its vertex at the median point on B, that faces the edge A. The angle is nothing but   θ  +      α    1  . Therefore   t  a  n  (  θ  +      α    1    )  =      A          B              /            2        =            2      A        B  Step 2Now, we may use a famous formula to write   t  a  n  (  θ  +      α    1    ) in a different way.  t  a  n  (  θ  +      α    1    )  =            t      a      n      (      θ      )      +      t      a      n      (              α        1            )              1      −      t      a      n      (      θ      )      t      a      n      (              α        1            )      Plugging in the value of   t  a  n  (  θ  ) and what we found for   t  a  n  (      α    1    ),we finally get  t  a  n  (  θ  +      α    1    )  =            2      ×      (      2      A      +      3      B      )              8      B      −      3      A      Put the same quantities equal to each other            2      A      +      3      B              8      B      −      3      A        =      A    B  A bit of simplification gives       A    2    +      B    2    −  2  A  B  =  0

eukrasicx · Accepted Answer

Step 1The length of one median is             a      2        +    4          b      2       and the other is       4          a      2        +          b      2      The intersection of the medians split each other with a ratio of 2:1Step 2Law of sines             sin      ⁡      x        a    =            sin      ⁡      y                      1        3                              a          2                +        4                  b          2                      sin  ⁡  x  =      3    5    sin  ⁡  y  =                    a        2            +      4              b        2                    5      a      Also   sin  ⁡  y  =      b          4              a        2            +              b        2                                a        2            +      4              b        2                    5      a        =      b          4              a        2            +              b        2                5  a  b  =      4          a      4        +    17          a      2              b      2        +    4          b      4          25      a    2        b    2    =  4      a    4    +  17      a    2        b    2    +  4      b    4      4      a    4    −  8      a    2        b    2    +      b    4    )  =  0    4  (      a    2    −      b    2        )    2    =  0  a  =  b you have an isoceles right triangle.

Find angles in a right triangle if tan theta=3/4, where theta is the angle between medians

Answered question

Answer & Explanation

New Questions in High school geometry