# 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.

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