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# M06-09

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Math Expert
Joined: 02 Sep 2009
Posts: 47977

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16 Sep 2014, 00:27
10
00:00

Difficulty:

65% (hard)

Question Stats:

65% (01:22) correct 35% (02:15) wrong based on 159 sessions

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What is the area of a triangle with the following vertices $$L(1, 3)$$, $$M(5, 1)$$, and $$N(3, 5)$$?

A. 3
B. 4
C. 5
D. 6
E. 7

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16 Sep 2014, 00:27
9
4
Official Solution:

What is the area of a triangle with the following vertices $$L(1, 3)$$, $$M(5, 1)$$, and $$N(3, 5)$$?

A. 3
B. 4
C. 5
D. 6
E. 7

Make a diagram:

Notice that the area of the blue square is $$4^2=16$$ and the area of the red triangle is 16 minus the areas of 3 little triangles which are in the corners ($$2*\frac{2}{2}$$, $$4*\frac{2}{2}$$ and $$4*\frac{2}{2}$$). Therefore, the area of a triangle LMN is $$16-(2+4+4)=6$$.

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24 Sep 2015, 12:05
a very elegant solution!

i used a longer approach: with Pythagorean theorem I found all the sides and that the traingle is an isosceles triangle with LM=MN, then found the height from M to base LN and finally the area
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Joined: 23 Oct 2015
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23 Oct 2015, 03:17
4
1
The question can be solved easily by using the area of the triangle by this Area of triangle=(1/2)|D|. where D is determinant.
|1 3 1|
|5 1 1|=1(1*1-5*1)-3(5*1-3*1)+1(5*5-3*1)=1(-4)-3(2)+1(25-3)=-4-6+22=12.
|3 5 1|

If you take positive value of determinant and then and half of that value is the area of triangle.

By this method we could solve the question in 35 odd seconds.
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23 Oct 2015, 09:56
1
viralashara wrote:
The question can be solved easily by using the area of the triangle by this Area of triangle=(1/2)|D|. where D is determinant.
|1 3 1|
|5 1 1|=1(1*1-5*1)-3(5*1-3*1)+1(5*5-3*1)=1(-4)-3(2)+1(25-3)=-4-6+22=12.
|3 5 1|

If you take positive value of determinant and then and half of that value is the area of triangle.

By this method we could solve the question in 35 odd seconds.

would u pls explain how did u get 1 in the right collumn?
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23 Oct 2015, 10:35
1
1
BelalHossain046 wrote:
viralashara wrote:
The question can be solved easily by using the area of the triangle by this Area of triangle=(1/2)|D|. where D is determinant.
|1 3 1|
|5 1 1|=1(1*1-5*1)-3(5*1-3*1)+1(5*5-3*1)=1(-4)-3(2)+1(25-3)=-4-6+22=12.
|3 5 1|

If you take positive value of determinant and then and half of that value is the area of triangle.

By this method we could solve the question in 35 odd seconds.

would u pls explain how did u get 1 in the right collumn?

Standard formula of Co-ordinate geometry.

If there are three points (a,b),(c,d) and (e,f)
The Area of triangle formed by these three points will be given by
1/2 *|a b 1|
|c d 1|
|e f 1|
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23 Oct 2015, 11:19
1. Calculate the equation of line connecting any 2 points (I used (1,3) and (3,5): Eq: x+2y=7)
2. Calculate the perpendicular distance from the 3rd point to the line (6/sq root (5))
3. Use formula for area of the triangle
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18 Mar 2016, 11:12
2
1
The area of triangle when 3 vertices are given by

= 1/2 [ x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]

x1,y1 = L(1,3 )
x2,y2 = M(5,1)
x3,y3 = N(3,5)

Substituting the 3 values in the formula we get

=> 1/2 [ 1 ( 1 -5 ) + 5( 5 - 3 ) + 3 ( 3 - 1 )]
= 1/2 [ -4 + 10 + 6 ] = 12/2 = 6

the tricky here is to remembering the correct sequence of the formula.
my short cut to remember this formula is to do a circular rotate first three vertices ( x1,y2,y3) by 1 to get second one ( x2,y3,y1) and do the rotate again to get the third one ( x3,y1,y2). OR you can form a 3x3 matrix to remember well.
1/2 [1,2,3 + 2,3,1 + 3,1,2].
Not sure if this is the wise approach but I could solve faster.
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17 Jun 2016, 10:21
1/2 *modulus of |x1-x2 x1-x3|
|y1-y2 y1-y3|
simple formula for finding area of triangle when 3 vertices (x1,y1);(x2,y2);(x3,y3) are given. | | indicates determinant.
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23 Jul 2016, 11:16
Can we do it like this ?

height will be 5 units as seen from the figure.

And the base will be LM which can be found by distance formula : 2\sqrt{5}

so area 1/2*5*2\sqrt{5} = 5\sqrt{5} (close to 6)

Is the approach right?
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30 Jul 2016, 11:53
Step 1: Calculate each side with distance formula

LM=2\sqrt{5} MN=2\sqrt{5} and LN= 2\sqrt{2}

Lets quickly use herons formula.

S= (2\sqrt{5}+\sqrt{5}+2\sqrt{2})/2

S=2\sqrt{5} +\sqrt{2}

Area of triangle= \sqrt{S(S-a)(S-b)(S-c)} = \sqrt{36} = 6
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07 Aug 2016, 19:23
Determinant approach, Brilliant solution
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28 Oct 2016, 21:12
1
Hey everyone,

This is really a new golden formula to me.

"The area of triangle when 3 vertices are given by

= 1/2 [ x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]"

Just have a concern that if x1 or y3 is negative, or y2-y3 negative, then we just apply the negative values into this formula?

Thanks alot
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20 Nov 2016, 05:55
I could solve it very quickly after I realised it is an isosceles triangle with sides sqrt(20), sqrt (20), sqrt (8). I simply used the formula to calculate the area of an isosceles triangle, which is [ a/4 * sqrt (4 * a^2 - c^2) ] such that side a = side b.

maybe it worked here, but i think the determinant approach, the square approach or the area of triangle using vertices approach are the most standard ones for solving such questions.
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03 May 2017, 07:24
Hi All,

We have to draw first the figure.

Then apply the formula : $$\frac{(Base*Height)}{2}$$
$$(LN*height)/2$$

Since LN is the combined length of two diagonals of a unit square, he have that LN = $$\sqrt{2}+\sqrt{2}$$
Since the Height is the combined length of three diagonals of a unit square we get : height : $$\sqrt{2}+\sqrt{2}+\sqrt{2}$$

So $$\frac{LN * height}{2}$$ = $$\frac{2\sqrt{2[}{square_root]*3[square_root]2}/2}$$

Area LMN = 6
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28 Jun 2017, 06:23
1
viralashara wrote:
The question can be solved easily by using the area of the triangle by this Area of triangle=(1/2)|D|. where D is determinant.
|1 3 1|
|5 1 1|=1(1*1-5*1)-3(5*1-3*1)+1(5*5-3*1)=1(-4)-3(2)+1(25-3)=-4-6+22=12.
|3 5 1|

If you take positive value of determinant and then and half of that value is the area of triangle.

By this method we could solve the question in 35 odd seconds.

Thank you for reminding this approach. I had totally forgotten about it. I'd like to add 1 more thing for everybody to understand..

Determinant of a matrix can be found by this general method..

|a b c|
|d e f |
|h i j|

= a*(e*j - i*f) - d*(b*j - I*c) + h*(b*f - e*c)

Note: You can take a, d, & h outside or a, b, & c; or b, e, & i; or c, f, & j; and so on i.e either a row or a column and subtract the cross-multiple of the rest as performed above.
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21 Nov 2017, 01:10
Clever approach.
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29 Dec 2017, 04:29
dharan wrote:
The area of triangle when 3 vertices are given by

= 1/2 [ x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]

x1,y1 = L(1,3 )
x2,y2 = M(5,1)
x3,y3 = N(3,5)

Substituting the 3 values in the formula we get

=> 1/2 [ 1 ( 1 -5 ) + 5( 5 - 3 ) + 3 ( 3 - 1 )]
= 1/2 [ -4 + 10 + 6 ] = 12/2 = 6

the tricky here is to remembering the correct sequence of the formula.
my short cut to remember this formula is to do a circular rotate first three vertices ( x1,y2,y3) by 1 to get second one ( x2,y3,y1) and do the rotate again to get the third one ( x3,y1,y2). OR you can form a 3x3 matrix to remember well.
1/2 [1,2,3 + 2,3,1 + 3,1,2].
Not sure if this is the wise approach but I could solve faster.

Hi could you clarify this.

In the formuar you have written $$\frac{1}{2}* [ x1 (y2- y3) + x2(y3-y1) + x3(y1-y2)$$

If the terms were interchanged like this $$\frac{1}{2} * [ x1(y2-y1) + x2( y1-y3) + x3(y3-y2) ]$$

Would it matter, what parameters govern this order.
Regards
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31 Dec 2017, 08:08
i solved using the traditional approach by finding the length of all sides. But great soln by Bunuel.
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20 May 2018, 10:50
I think this is a high-quality question and I agree with explanation.
Re M06-09 &nbs [#permalink] 20 May 2018, 10:50

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