In Fig. 7.9, ABC is a right triangle and right angled at B such that ∠BCA = 2 ∠BAC. Show that hypotenuse AC = 2 BC.
Solution:
Given, ABC is a right triangle.
The triangle is right angled at B such that ∠BCA = 2 ∠BAC
We have to show that the hypotenuse AC = 2 BC.
Extend CB to a point D such that BC = BD
Let ∠BAC = x
So, ∠BCA = 2x
Considering triangles ABC and ABD,
BC = BD ------------------ (1)
AB = common side
∠ABC = ∠ABD = 90
SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are congruent.
By SAS criterion, the triangles ABC and ABD are congruent.
Considering triangles ABC and ABD,
AD = AC ------------------- (2)
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 CPCT, ∠DAB = ∠BAC
∠DAB = x
From the figure,
∠DAC = ∠DAB + ∠BAC
∠DAC = x + x
∠DAC = 2x
So, ∠DAC = ∠ACD
We know that the sides opposite to equal angles are equal.
DC = AD
From (2), AC = DC
From (1), AC = BC + BC
Therefore, AC = 2BC
✦ Try This: In a ΔABC, AD is the bisector of ∠BAC If AB = 6 cm, AC = 5 cm and BD = 3 cm„ then DC =
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.4 Sample Problem 1
In Fig. 7.9, ABC is a right triangle and right angled at B such that ∠BCA = 2 ∠BAC. Show that hypotenuse AC = 2 BC
Summary:
A right-angled triangle is a type of triangle that has one of its angles equal to 90 degrees. The other two angles sum up to 90 degrees. In Fig. 7.9, ABC is a right triangle and right angled at B such that ∠BCA = 2 ∠BAC. It is shown that hypotenuse AC = 2 BC
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