# ABC is an isosceles triangle with AC = BC. If AB^{2} = 2AC^{2}, prove that ABC is a right triangle

**Solution:**

We know that, in a triangle, if the square of one side is equal to the sum of the squares of the other two sides then the angle opposite the first side is a right angle.

In ΔABC,

It is given that AC = BC and AB^{2} = 2 AC^{2}

⇒ AB^{2} = AC^{2} + AC^{2}

⇒ AB^{2} = AC^{2} + BC^{2} [Since AC = BC]

As the above equation satisfies Pythagoras theorem, we can say that

⇒ ∠ACB = 90°

Therefore, ΔABC is a right triangle.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 6

**Video Solution:**

## ABC is an isosceles triangle with AC = BC. If AB² = 2AC², prove that ABC is a right triangle

NCERT Class 10 Maths Solutions Chapter 6 Exercise 6.5 Question 5

**Summary:**

If ABC is an isosceles triangle with AC = BC and if AB^{2} = 2AC^{2}, it is proved that ABC is a right triangle.

**☛ Related Questions:**

- ABC is an equilateral triangle of side 2a. Find each of its altitudes.
- Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
- In Figure 6.54, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that(i) OA2 + OB2 + OC2 - OD2 - OE2 - OF2 = AF2 + BD2 + CE2(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2
- A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

Math worksheets and

visual curriculum

visual curriculum