Jump to content
BrainDen.com - Brain Teasers
  • 0

Reflect on this



10 answers to this question

Recommended Posts

  • 0

To get from (7,1) to (10,5) you need to take 3 steps right and 4 steps up,

so a straight line through these points croses line x=8 4/3 units above (7,1) (since it's 1 step right).
Therefore the coordinates of 4th vertex of the quadrilateral are (8, 1+4/3).
Given this caclulation of area of the quadrilateral is elementary: 4*(8-(2+2/3))/2 = 2*(5+1/3) = 10+2/3.

Link to comment
Share on other sites

  • 0

spoiler alert

Flipping about line x creates 2 more points for a 5 point polygon (since we are only interested in total area and not the overlapping area). The (8,-3) point remains the same after the flip. The two new points are (7,1) and (10,5). We can simplify this 5 point polygon into a normal triangle by realizing that the points (7,1) and (9,1) lie exactly along the lines created by the points along y=5 and y=-3. Now just solve for that area.

Simplify further by splitting that triangle into 2 right angle triangles with a new point at (8,5)

a=1/2 bh, and since there are two, the total area is base times height. b=4, h=8, area = 32.

Link to comment
Share on other sites

  • 0

No. It should be included, but only once.

The two triangles (the original and its mirror image) partially overlap,

making a four-sided figure with vertices at (6,5) (8,-3) (10,5) and a point

that is the intersection of lines segments from (6,5) to (9,1) and from (10,5) to (7,1).

We want the area of that four-sided figure.

Link to comment
Share on other sites

  • 0

Lets say the area of the triangle is A

The area of reflected triangle in also A

Lets say area of overlap is O

The area we are looking for is 2A - O

Area O is a kite with diagonals 2 and 5 (haven't drawn it but only pictured it... so not sure if this is correct)

Therefore area (d1d2/2) of O = 5

The area we want is 2A - 5

Now for the aha moment:

2A - 5 = A + 1/4A = 5/4A

3A/4 = 5

A = 20/3

So the area we want (2A - O) = 25/3

Link to comment
Share on other sites

  • 0


The interesting point of intersection is on x=8 line. The y coordinate of that point is 7/3. Due to the symmetry along the x=8 line we can only consider the are of the left side and multiply it by 2. The area of that triangle is half of the area of a rectangle 16/3 x 2. So the area we're looking for is the area of that rectangle and is 32/3

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.


  • Recently Browsing   0 members

    • No registered users viewing this page.
  • Create New...