ABC is a right triangle such that AB = AC and bisector of angle C intersects the side AB at D. Prove that AC + AD = BC.
Solution:
Given, ABC is right triangle
AB = AC
Bisector of angle C intersects the side AB at D.
We have to prove that AC + AD = BC
In right triangle ABC,
AB = AC
BC is the hypotenuse
∠A = 90°
Draw DE perpendicular to BC
∠E = 90°
∠3 = ∠4 = 90°
Considering triangles DAC and DEC,
∠A = ∠3 = 90°
CD is the bisector of angle C.
So, ∠1 = ∠2
Common side = CD
AAS criterion (Angle-angle-side) states that when two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.
By AAS criterion, the triangles DAC and DEC are congruent.
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCTC,
DA = DE ------------ (1)
AC = EC ------------ (2)
In triangle ABC,
AB = AC
We know that the angles opposite to equal sides are equal.
So, ∠C = ∠B ------------- (3)
By angle sum property,
∠A + ∠B + ∠C = 180°
From (3), ∠A + ∠B + ∠B = 180°
∠A + 2∠B = 180°
90° + 2∠B = 180°
2∠B = 180° - 90°
∠B = 90°/2
∠B = 45°
In triangle BED,
∠B + ∠E + ∠D = 180°
∠B + ∠4 + ∠5 = 180°
90° + ∠5 = 180° - ∠B
∠5 = 180° - 45° - 90°
∠5 = 45°
So, ∠B = ∠5
We know that the sides opposite to equal angles are equal.
DE = BE -------------- (4)
From (1) and (4),
DA = DE = BE ----------- (5)
From the figure,
BC = BE + CE
From (2) and (5),
BC = CA + DA
Therefore, BC = AC + AD
✦ Try This: The diagonal BD of a parallelogram ABCD bisects angles B and D. Prove that ABCD is a rhombus.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 18
ABC is a right triangle such that AB = AC and bisector of angle C intersects the side AB at D. Prove that AC + AD = BC
Summary:
ABC is a right triangle such that AB = AC and bisector of angle C intersects the side AB at D. It is proven that AC + AD = BC by angle sum property which states that the sum of all three interior angles of a triangle is always equal to 180 degrees
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