Question

The point (2,-2),(-1,2),(3,5) are the vertices of ................ the triangle.
a) equilateral triangle
b) isosceles triangle
c)right angled triangle
d) right angled isosceles triangle

pls help me fastly i ll mark it as brainlist answer......​

Answer 1

Step-by-step explanation:

pls mark as the brainlest

Answer 2

The points (2, -2), (-1, 2), and (3, 5) are the vertices of a right-angled isosceles triangle.

Step-by-step Explanation

Given:

The points (2, -2), (-1, 2), and (3, 5) are the vertices of the given triangle.

To be found:

To find the type of triangle formed by the given points.

Formula to be used:

• The distance between two points is calculated using the formula, $Distance=\sqrt{(x_{2}-x_{1} )^{2}+ (y_{2}-y_{1} )^{2}}$
• According to Pythagoras' Theorem, $c=\sqrt{a^{2} +b^{2} }$, where $a$ and $b$ are sides of a right-angled triangle and $c$ is the hypotenuse.

Let the vertices of the given triangle be A(2, -2), B(-1, 2), and C(3, 5).

Now, measuring the side AB, we get

$AB=\sqrt{(-1-2 )^{2}+ (2-(-2) )^{2}} =\sqrt{(-3)^{2}+4^{2} } \\\implies AB=\sqrt{9+16}=\sqrt{25}$

$\implies AB =5$ units

Measuring the side BC, we get

$BC=\sqrt{(3-(-1) )^{2}+ (5-2 )^{2}} =\sqrt{4^{2}+3^{2} } \\\implies BC=\sqrt{16+9}=\sqrt{25}$

$\implies BC =5$ units

Measuring the side AC, we get

$AC=\sqrt{(3-2)^{2}+ (5-(-2) )^{2}} =\sqrt{1^{2}+7^{2} } \\\implies AC=\sqrt{1+49}=\sqrt{50}$

$\implies AC =5\sqrt{2}$ units

Here, we get to know that, $AB=BC = 5$ units.

Since the two sides of the triangle are equal, it is an isosceles triangle.

Next, to check whether the given isosceles triangle is right-angled isosceles triangle, we use the Pythagoras Theorem.

$AC=\sqrt{AB^{2} +BC^{2} } \\\implies 5\sqrt{2} =\sqrt{5^{2} +5^{2} }=\sqrt{25+25} \\\implies 5\sqrt{2} = \sqrt{50}$

Hence, it is confirmed that the given isosceles triangle is a right-angled triangle.

Therefore, using the distance formula and Pythagoras' theorem, we found that the points (2, -2), (-1, 2), and (3, 5) are the vertices of a (d) right-angled isosceles triangle.

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