In ΔABC, AD is the perpendicular bisector of BC (see Fig. 7.30). Show that ΔABC is an isosceles triangle in which AB = AC.

Solution:

Given: AD is the perpendicular bisector of BC means ∠ADB = ∠ADC = 90° and BD = DC

To Prove: ΔABC is an isosceles triangle in which AB = AC.

In ΔADC and ΔADB,

AD = AD (Common)

∠ADC = ∠ADB (Each 90°)

CD = BD (AD is the perpendicular bisector of BC)

∴ ΔADC ≅ ΔADB (By SAS congruence rule)

∴ AB = AC (By CPCT)

Therefore, ABC is an isosceles triangle in which AB = AC.

☛ Check: NCERT Solutions Class 9 Maths Chapter 7

Video Solution:

In ΔABC, AD is the perpendicular bisector of BC (see Fig. 7.30). Show that ΔABC is an isosceles triangle in which AB = AC.

NCERT Maths Solutions Class 9 Chapter 7 Exercise 7.2 Question 2

Summary:

If in ΔABC, AD is the perpendicular bisector of BC, then we have proved that ΔABC is an isosceles triangle in which AB = AC.

☛ Related Questions:

• ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig. 7.31). Show that these altitudes are equal.
• ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig. 7.32). Show that(i) ΔABE ≅ ΔACF(ii) AB = AC, i.e., ABC is an isosceles triangle.
• ABC and DBC are two isosceles triangles on the same base BC (see Fig. 7.33). Show that ∠ABD = ∠ACD.
• ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that ∠BCD is a right angle.