Question

If ABCD is a rectangle, calculate x as a function of α

Answer 1

Explanation:

The length of this triangle is 10 squares

and the width is 4 squares

The diagonals divide the rectangle into four triangles

These traingles are isoceles

Each two triangles facing each others are identical

<B = 90 degree

B = alpha + Beta

Let Beta be the angle next alpha

The segment that is crossing Beta is its bisector since it perpendicular to the diagonals wich means that:

Beta = 2x

Then B = alpha + 2x

90 = alpha +2x

90-alpha = 2x

x = (90-alpha)/2

Answer 2

x = 90 - 2α

Explanation:

Solution:-

- Consider the right angled triangle " ABD ". The sum of angles of an triangle is always "180°".

< BAD > + < ADB > + < ABD > = 180°

< ABD > = 180 - 90° - α

< ABD > = 90° - α

- Then we look at the figure for the triangle "ABE". Where " E " is the midpoint and intersection point of two diagonals " AC and BD ".

- We name the foot of the perpendicular bisector as " F ": " BF " would be the perpendicular bisector. The angle < BAE > is equal to < ABD >.

< ABD > = < BAE > = 90° - α ... ( Isosceles triangle " BEA " )

Where, sides ( BE = AE ).

- Use the law of sum of angles in a triangle and consider the triangle " BFA " as follows:

< ABF> + < BFA > + < BAF > = 180°

< ABF > = 180 - (90° - α) - 90°

< ABF > = α

Where, < BAF > = < BAE >

- The angle < ABD > = < ABE > is comprised of two angles namely, < ABF > and < FBE > = x.

< ABD > = < ABE > = < ABF > + x

90° - α = α + x

x = 90 - 2α ... Answer