Q4 of 23 Page 220

The vertices of a triangle are the points with coordinates (3, 5), (9, 13), (10, 6). Prove that the triangle is isosceles. Calculate is area.

The three vertices are (3, 5), (9, 13), (10, 6)


Length of first side = √((9 - 3)2 + (13 - 5)2)


⇒ Length of first side = √(62 + 82)


⇒ Length of first side = 10 units


Length of second side = √((10 - 9)2 + (6 - 13)2)


⇒ Length of second side = √(12 + 72)


⇒ Length of second side = √50 units


Length of third side = √((10 - 3)2 + (6 - 5)2)


⇒ Length of third side = √(72 + 12)


⇒ Length of third side = √50 units


Since the length of second and third sides are equal so the triangle is isosceles


Length of height


Length of height = √75 units


Area = 0.5 × base × height


⇒ Area = 12.5 × 1.732


⇒ Area = 12.5 × 1.732 = 10.768 sq. units


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2

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i) Find the coordinates of the midpoints of all sides.


ii) Prove that the quadrilateral with these midpoints as vertices is a parallelogram.

 3

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i) Find the coordinates of the fourth vertex of the small quadrilateral.


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