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# Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles triangle

**Solution:**

An isosceles triangle is a triangle that has two sides of equal length.

To check whether the given points are vertices of an isosceles triangle, we need to check the distance between any of the 2 points should be the same for two pairs of given points.

Let the points (5, - 2), (6, 4), and (7, - 2) represent the vertices A, B, and C of the given triangle

We know that the distance between the two points is given by the Distance Formula,

Distance Formula = √[(x₂_{ - }x₁)^{2} + (y₂ - y₁)^{2}]

To find AB, that is distance between points A (5, - 2) and B (6, 4), let x₁ = 5, y₁ = -2, x₂ = 6, y₂ = 4

AB = √[( 6 - 5_{ })^{2} + (4 - (-2))^{2}]

= √[(1)^{2} + (6)^{2}]

= √1 + 36

= √37

To find BC, distance between Points B (6, 4) and C (7, - 2), let x₁ = 6, y₁ = 4, x₂ = 7, y₂ = - 2

BC = √ [( 7 - 6_{ })^{2} + (-2 - 4)^{2}]

= √[(1)^{2} + (- 6)^{2}]

= √1 + 36

= √37

To find AC, that is distance between Points A (5, - 2) and C (7, - 2), let x₁ = 5, y₁ = - 2, x₂ = 7, y₂ = - 2

AC = √ [( 7 - 5_{ })^{2} + (-2 - (- 2))^{2}]

= √[(2)^{2} + (0)^{2}]

= 2

From the above values of AB, BC and AC we can conclude that AB = BC. As the two sides are equal in length, therefore, ABC is an isosceles triangle.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 7

**Video Solution:**

## Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles triangle

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.1 Question 4

**Summary:**

The points (5, - 2), (6, 4), and (7, - 2) are the vertices of an isosceles triangle.

**☛ Related Questions:**

- In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.
- Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:(i) (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0) (ii) (- 3, 5), (3, 1), (0, 3), (- 1, - 4) (iii) (4, 5), (7, 6), (4, 3), (1, 2)
- Find the point on the x-axis which is equidistant from (2, - 5) and (- 2, 9).
- Find the values of y for which the distance between the points P (2, - 3) and Q (10, y) is 10 units.

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