If in a quadrilateral abcd ,ac bisects angle a and c show that ac is perpendicular to bd
pallavi butte , 7 Years ago
Grade 9
Analytical Geometry> If in a quadrilateral abcd ,ac bisects an...
4 AnswersPranjal Animesh
Last Activity: 7 Years ago
In quadrilateral ABCD ,
AC and BD are diagonals . So let they intersect at O.
In triangles ABC and ADC,
angle CAB = angle CAD
angle ACB = angle ACD
AC = AC
Therefore, triangles ABC and ADC are congruent.
NOW,
AB = AD (corresponding parts of congruent triangles)
In triangle ABD,
AB = AD
So, triangle ABD is isosceles.
As we know the angle bisector of the vertical angle of an isosceles triangle
is also the perpendicular bisector of the base.
So AC intersects BD at right angle.
so angle AOD = 90o.
Pranjal Animesh
Last Activity: 7 Years ago
In quadrilateral ABCD ,
AC and BD are diagonals . So let they intersect at O.
In triangles ABC and ADC,
angle CAB = angle CAD
angle ACB = angle ACD
AC = AC
Therefore, triangles ABC and ADC are congruent.
NOW,
AB = AD (corresponding parts of congruent triangles)
In triangle ABD,
AB = AD
So, triangle ABD is isosceles.
As we know the angle bisector of the vertical angle of an isosceles triangle
is also the perpendicular bisector of the base.
So AC intersects BD at right angle
Ayan
Last Activity: 4 Years ago
In triangles ABC and ADC,
angle CAB = angle CAD
angle ACB = angle ACD
AC = AC
Therefore, triangles ABC and ADC are congruent.
NOW,
AB = AD (corresponding parts of congruent triangles)
In triangle ABD,
AB = AD
So, triangle ABD is isosceles.
As we know the angle bisector of the vertical angle of an isosceles triangle
is also the perpendicular bisector of the base.
So AC intersects BD at right angle
Ram Kushwah
Last Activity: 4 Years ago
Let the diagonals AC and BD intersect at O
In ΔABC and Δ ACD
∠DAC= ∠BAC
∠ACD= ∠BCA
AC is common
ΔABC≅ Δ ACD
So AB=AD ( side opposite to equal angles)
Now In ΔAOB and Δ AOD
AB=AD ( proved)
∠DAC= ∠BAC ( Given)
AO is common
ΔAOB ≅ Δ AOD
Thus ∠AOB= ∠AOD ( angles opposite to equal sides)
But ∠AOB are ∠AOD adjacent angles
So ∠AOB+∠AOD=180°
∠AOB+∠AOB=180°
So ∠AOB=∠AOD=90°
Hence AC is perpendicular to BD
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