Online Notes and Lectures: Geometry Problems

Showing posts with label Geometry Problems. Show all posts

Thursday, January 9, 2014

Find the centroid of a triangle whose vertices are formed by (2, 7),(3, -1),(-5, 6)

0 comments Posted by Anonymous at 12:04 AM

Hits:

Problem: How to find the centroid of a triangle whose vertices are formed by (2, 7),(3, -1),(-5, 6)

$(2,7),(3,-1),(-5,6)\,$

Solutions:

The formula for finding the centroid or median of a triangle whose vertices are

$(x_1,y_1),(x_2,y_2),(x_3,y_3)\,$ is $\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\,$

Applying the formula, we get

$\left(\frac{2+3-5}{3},\frac{7-1+6}{3}\right)\,$

$\left (0,4\right)\,$

Therefore, the centroid of the triangle formed is at P(0, 4).

List of Similar Problems with Complete Solutions

How to find the centroid of a polygon
How to find distance between two points
Examples Problems with Solutions in Geometry

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: centroid, Geometry Problems

Wednesday, January 8, 2014

Find the value of x if the area of the triangle formed by (10, 2),(-3, -4),(x, 1), is 5

0 comments Posted by Anonymous at 11:49 PM

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Problem: How to find the value of x if the area of the triangle formed by (10, 2),(-3, -4),(x, 1), is 5 square units

$(10,2),(-3,-4),(x,1)\,$

Solutions:

The area of triangle is

$\frac{1}{2}|[10(-4-1)-3(1-2)+x(2+4)]|=5\,$

$\frac{1}{2}|-50+3+6x|=5\,$

$6x-47=10\,$

$x=\frac{57}{6}\,$

$x=\frac{19}{2}\,$

Therefore, the value of x is 19/2 units.

List of Similar Problems with Complete Solutions

How to find the area of a polygon
How to find distance between two points
Examples Problems with Solutions in Geometry

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: Area, Geometry Problems

Thursday, January 2, 2014

Find the area of the triangle formed by (2, 4),(2, 6),(2+sqrt{3}, 5)

0 comments Posted by Anonymous at 8:38 PM

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Problem: How to find the area of the triangle formed by the points (2, 4),(2, 6),(2+sqrt{3}, 5)

$(2,4),(2,6),(2+\sqrt{3},5)\,$

Solutions:

The formula for finding the area of the triangle formed by

$(x_1,y_1),(x_2,y_2),(x_3,y_3)\,$ is

$\frac{1}{2}|[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]|\,$

Applying the formula,we get

$\frac{1}{2}|[2(6-5)+2(5-4)+(2+\sqrt{3})(4-6)]|\,$

$\frac{1}{2}|2+2-4-2\sqrt{3}|\,$

$\frac{1}{2}|-2\sqrt{3}|\,$

Therefore, the area of the triangle is $\sqrt{3}\,$ .

List of Similar Problems with Complete Solutions

How to find the area of a polygon
How to find distance between two points
Examples Problems with Solutions in Geometry

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: Area, Geometry Problems

Find the area of the triangle formed by (1, 3),(3, -1),(-5, -5)

0 comments Posted by Anonymous at 8:25 PM

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Problem: How to find the area of the triangle formed by the points (1, 3),(3, -1),(-5, -5)

$(1,3),(3,-1),(-5,-5)\,$

Solutions:

The formula for finding the area of the triangle formed by

$(x_1,y_1),(x_2,y_2),(x_3,y_3)\,$ is

$\frac{1}{2}|[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]|\,$

Applying the formula,we get

$\frac{1}{2}|[1(-1+5)+3(-5-3)-5(3+1)]|\,$

$\frac{1}{2}|4-24-20|\,$

$\frac{1}{2}|-40|\,$

Therefore, the area of the triangle is 20.

List of Similar Problems with Complete Solutions

How to find the area of a polygon
How to find distance between two points
Examples Problems with Solutions in Geometry

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: Area, Geometry Problems

Find the area of the triangle formed by (5, 2),(-9, -3),(-3, -5)

0 comments Posted by Anonymous at 12:13 AM

Hits:

How to find the area of the triangle formed by the points (5, 2),(-9, -3),(-3, -5).

$(5,2),(-9,-3),(-3,-5)\,$

Solutions:

The formula for finding the area of the triangle formed by

$(x_1,y_1),(x_2,y_2),(x_3,y_3)\,$ is

$\frac{1}{2}|[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]|\,$

Applying the formula,we get

$\frac{1}{2}|[5(-3+5)-9(-5-2)-3(2+3)]|\,$

$\frac{1}{2}|10+63-15|\,$

$\frac{1}{2}|58|\,$

Therefore, the area of the triangle is 29.

List of Similar Problems with Complete Solutions

How to find the area of a polygon
How to find distance between two points
Examples Problems with Solutions in Geometry

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: Area, Geometry Problems

Tuesday, December 31, 2013

Find the relation between x and y, if the point (x, y) is equidistant from (7, -6),(-3, 4)

0 comments Posted by Anonymous at 4:18 PM

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Problem: How to Find the relation between x and y, if the point (x, y)is equidistant from (7, -6),(-3, 4).

$(7,-6),(-3,4)\,$

Solutions:

Find the distance between (x,y) and the first point is

$\sqrt{(x-7)^2+(y+6)^2}\,$

$\sqrt{x^2+49-14x+y^2+36+12y}\,$

$\sqrt{x^2+y^2-14x+12y+85}\,$

Let this expression be 1.

Find the distance between (x,y) and the second point is

$\sqrt{(x+3)^2+(y-4)^2}\,$

$\sqrt{x^2+9+6x+y^2+16-8y}\,$

$\sqrt{x^2+y^2+6x-8y+25}\,$

Let this expression be 2.

As per the given condition,equating both the expressions,

$\sqrt{x^2+y^2-14x-12y+85}=\sqrt{x^2+y^2+6x-8y+25}\,$

Squaring on both sides,we get

$x^2+y^2-14x+12y+22=x^2+y^2+6x-8y+25\,$

cancelling the terms which are common on both sides,

$-14x-6x+12y+8y=-60\,$

$-20x+20y=-60\,$

List of Similar Problems with Complete Solutions

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: distance, Geometry Problems

Find the value of a if the distance between the points (a, 2),(3, 4) is 2

0 comments Posted by Anonymous at 3:34 PM

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Problem: How to find the value of a if the distance between the points (a, 2),(3, 4) is 2.

$(a,2),(3,4)\,$

Solutions:

The distance between the given points is

$\sqrt{(a-3)^2+(2-4)^2}=2\,$

Simplifying

$\sqrt{a^2+9-6a+4}=2\,$

Squaring on both sides,

$a^2-6a+13=4\,$

$a^2-6a+9=0\,$

$(a-3)^2=0\,$

Finally, the value of a is 3.

List of Similar Problems with Complete Solutions

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: distance, Geometry Problems

Saturday, December 28, 2013

Given (0, 9) and (6, -9), find the midpoint, distance, slope, and equation of the line.

0 comments Posted by Anonymous at 11:12 PM

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Problem: Given these pairs of points, (0, 9)) and (6, -9), find the midpoint, distance, slope, and equation of the line.

$(0,9),(6,-9)\,$

Solutions:

To find the midpoint, average the x coordinates and y coordinates. The midpoint is

$\left(\frac{0+6}{2},\frac{9-9}{2}\right) = (3,0)\,$

To find the (always zero or positive) distance, use the formula

$d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,$

$d = \sqrt{(6)^2+(-9-9)^2} = \sqrt{36+18^2} = \sqrt{(2\cdot 3)^2+(2\cdot 3^2)^2} = \sqrt{(2\cdot 3)^2(1+3^2} = 6\sqrt{10}\,$

To find the slope, use the formula

$m = \frac{y_2-y_1}{x_2-x_1}\,$

$m = \frac{-9-9}{6-0} = -3\,$

The equations of the line are

Method 1:

$y=mx+b\,$

Plug in one known point (say, (0, 9)) and the calculated slope.

$9 = -3\cdot 0 + b\,$

$b = 9\,$

Now plug b and m into the line equation:
$y = -3x + 9\,$

Method 2:

$(y-y_1) = m(x-x_1)\,$

Plug in one known point (say, (6, –9) ) and the calculated slope.

$(y+9) = -3(x-6)\,$

$y = -3x + 18 - 9\,$

$y = -3x + 9\,$

List of Similar Problems with Complete Solutions

NOTE:

For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

Tags: distance, equation of line, Geometry Problems, midpoint, slope

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Thursday, January 9, 2014

Problem: How to find the centroid of a triangle whose vertices are formed by (2, 7),(3, -1),(-5, 6)

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Wednesday, January 8, 2014

Problem: How to find the value of x if the area of the triangle formed by (10, 2),(-3, -4),(x, 1), is 5 square units

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Thursday, January 2, 2014

Problem: How to find the area of the triangle formed by the points (2, 4),(2, 6),(2+sqrt{3}, 5)

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Problem: How to find the area of the triangle formed by the points (1, 3),(3, -1),(-5, -5)

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

How to find the area of the triangle formed by the points (5, 2),(-9, -3),(-3, -5).

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Tuesday, December 31, 2013

Problem: How to Find the relation between x and y, if the point (x, y)is equidistant from (7, -6),(-3, 4).

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Problem: How to find the value of a if the distance between the points (a, 2),(3, 4) is 2.

Solutions:

Simplifying

List of Similar Problems with Complete Solutions

NOTE:

Saturday, December 28, 2013

Problem: Given these pairs of points, (0, 9)) and (6, -9), find the midpoint, distance, slope, and equation of the line.

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

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