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Question

Prove that the points (–3, 0), (1, –3) and (4, 1) are the vertices of an isosceles right angled triangle. Find the area of this triangle.
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Solution

Formula: 1 Mark
Concept of right-angled triangles: 1 Mark
Steps: 1 Mark
Answer: 1 Mark

Let A(-3, 0), B(1, -3) and C(4, 1) be the given points. Then,

Distance between the points is given by
(x1x2)2+(y1y2)2

AB=(1(3))2+(30)2=16+9=5 units.

BC=(41)2+(1+3)2=9+16=5 units.

AC=(4+3)2+(10)2=49+1=52 units.

Clearly, AB = BC. Therefore, ΔABC is isosceles.

Also,

AB2+BC2=25+25=(52)2=CA2

ΔABC is right- angled at B because it is satisfying pythagoras theorem.

Thus, ΔABC is right- angled isosceles triangle.

Now, Area of ΔABC=12(Base×Height)

=12(AB×BC)

Area of ΔABC=(12×AB×BC) sq. units

=252 sq. units.


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