Question

show that 7, 10 - 2, 5 and 3, - 4 and vertices of right angle isosceles triangle

Answer 1

Question:

Show that the points (7, 10), (-2, 5) and (3, -4) are the vertices of a right angled isosceles triangle.

Answer:

The sides of a right angled isosceles triangle are always in the ratio 1 : 1 : √2.

We have to find the distances between each two points to get the length of sides of the triangle.

Distance between points (7, 10) and (-2, 5)

⇒  √((7 - (-2))² + (10 - 5)²)

⇒  √(9² + 5²)

⇒  √(81 + 25)

⇒  √106

Distance between points (-2, 5) and (3, -4)

⇒  √((- 2 - 3)² + (5 - (-4))²)

⇒  √((- 5)² + 9²)

⇒  √(25 + 81)

⇒  √106

Distance between points (7, 10) and (3, -4)

⇒  √((7 - 3)² + (10 - (-4))²)

⇒  √(4² + 14²)

⇒  √(16 + 196)

⇒  √212

⇒  √106 × √2

Thus the lengths of the sides of the triangle are √106 units, √106 units and √212 units.

√106 : √106 : √212  =  1 : 1 : √2

As the lengths of sides of the triangle keeps the same ratio as that mentioned at first, we can say that the triangle is right angled isosceles triangle.

Hence Proved!!!