Question

Name the type of the quadrilateral formed by the points (1, 7), (4, 2), (—1, —1) and (—4, 4).

Answer 1

the quadrilateral is a sqaure as it sides and diagonals are equal...

pls refer my attachment

hope it helps...

Answer 2

The type of quadrilateral formed by the points (1, 7), (4, 2), (-1, -1), and (-4, 4) is Square.

Concept used:

We use Distance Formula to find the length of the sides and diagonals.

The distance between two points $A\left(x_{1}, y_{1}\right) \text { and } B\left(x_{2}, y_{2}\right)$ is

AB =  √( difference of abscissae ) ² + ( difference of ordinates ) ²

$A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Types of quadrilaterals and their conditions:

Parallelogram:

• In a parallelogram both pairs of opposite sides are equal.
• Diagonals bisect each other.

Rectangle:

• In a rectangle diagonals are equal.
• Diagonals bisect each other.
• Opposite sides are equal.

Rhombus:

• In Rhombus all the four sides are equal.
• Diagonals bisect each other at a right angle.
• Diagonals are not equal.

Square:

• In a square all four sides are equal.
• Diagonals are equal.
• Diagonals bisect each other at a right angle.

Given :

Let A (1, 7), B (4, 2), C (-1, -1) and D (- 4, 4) .

To Find:

The type of the quadrilateral formed by the given points.

Solution :

Step 1: Find the length of side AB by using the distance formula :

$\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Vertices :

$\begin{array}{l}A\left(x_{1}=1, y_{1}=7\right), B\left(x_{2}=4, y_{2}=2\right) \\\\\ \text{ Length of side } A B=\sqrt{(4-1)^{2}+(2-7)^{2}} \\\\AB=\sqrt{3^{2}+(-5)^{2}} \\\\A B=\sqrt{9+25} \\\\AB=\sqrt{34 \text { units }}\end{array}$

Step 2: Find the length of side BC by using the distance formula :

$\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Vertices :

$B\left(x_{1}=4, y_{1}=2\right), C\left(x_{2}=-1, y_{2}=-1\right)$

$\text { Length of side } B C=\sqrt{(-1-4)^{2}+(-2-1)^{2}}$

$\begin{array}{l}B C=\sqrt{(-5)^{2}+(-3)^{2}} \\\\\ B C=\sqrt{25+9} \\\\\ B C=\sqrt{34} \text { units }\end{array}$

Step 3: Find the length of the side CD by using the distance formula :

$\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Vertices :

$C\left(x_{1}=-1, y_{1}=-1\right) \text { and } D\left(x_{2}=-4, y_{2}=4\right)$

$\text { Length of side } C D=\sqrt{(-4+1)^{2}+(4+1)^{2}}$

$\begin{array}{l}C D=\sqrt{(-3)^{2}+(5)^{2}} \\\\\ C D=\sqrt{9+25} \\\\\ C D=\sqrt{34} \text { units }\end{array}$

Step 4: Find the length of side DA by using the distance formula :

$\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Vertices :

$D\left(x_{1}=-4, y_{1}=4\right), A\left(x_{2}=1, y_{2}=7\right)$

$\text { Length of side DA }=\sqrt{(-4-1)^{2}+(4-7)^{2}}$

$\begin{array}{l}D A=\sqrt{(-5)^{2}+3^{2}} \\\\\ D A=\sqrt{25+9} \\\\\ D A=\sqrt{34} \text { units }\end{array}$

Step 5: Find the length of diagonal BD by using the distance formula :

$\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Vertices :

$B\left(x_{1}=4, y_{1}=2\right), D\left(x_{2}=-4, y_{2}=4\right)$

$\text { Length of diagonal } B D=\sqrt{(-4-4)^{2}+(4-2)^{2}}$

$\begin{array}{l}B D=\sqrt{(-8)^{2}+2^{2}} \\\\\ BD=\sqrt{64+4} \\\\\ B D=\sqrt{68} \text { units }\end{array}$

Step 6: Find the length of  diagonal AC by using the distance formula :

$\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

Vertices :

$A\left(x_{1}=1, y_{1}=7\right), C\left(x_{2}=-1, y_{2}=-1\right)$

$\text { Length of diagonal } A C=\sqrt{(-1-1)^{2}+(-1-7)^{2}}$

$\begin{array}{l}A C=\sqrt{(-2)^{2}+(-8)^{2}} \\\\\ A C=\sqrt{4+64} \\\\\ A C=\sqrt{68} \text { units }\end{array}$

Since all the four sides (AB = BC = CD = DA = √34)  and diagonal  (BD = AC = √68) both are equal.

Hence, the given points are the vertices of a square.

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Question 6 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)

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