What is the area of the shaded region?

2 years ago by zkfcfbzr to c/dailymaths

An 8x5 rectangle. If the bottom left corner is considered (0, 0), then two lines are drawn within the rectangle, from (0, 4) to (8, 1) and from (1, 5) to (7, 0). The smaller two regions of the four these lines cut the rectangle into are shaded. What is their combined area?

clara 16 points 2 years ago

this was my approach

the same 8x5 rectangle, with additional lines added

detailed annotation for screen readers here: 4 lines have been added to the original depiction. the first set of 2 lines make a triangle shape using coordinates (4, 2.5) to (4, 0) to (7, 0). the second set of 2 lines make a triangle shape using coordinates (4, 2.5) to (8, 2.5) to (8, 1)

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WhatThaFudge 7 points 2 years ago

spoiler

( (4x2.5)- (((2.5x3)/2)+((4x1.5)/2)) )x2 = 6.5

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graham1 4 points 2 years ago

That's what I did too

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crt0o 9 points 2 years ago

It's 6.5

| 0  5  1 |   | 0   4   1 |
| 0  4  1 | + | 4  2.5  1 | = 6.5
| 1  5  1 |   | 1   5   1 |

Determinants go brrr

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morphballganon 8 points 2 years ago

6.5 square units

The description doesn't match the image. The image shows two lines that intersect. The description's lines would not intersect as stated. My answer is for the image, not the description.

The first line's coordinates should be 0,4 to 8,1. The second should be 1,5 to 7,0.

I see you corrected it.

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TootSweet 6 points 2 years ago

Huh. I got 7.5 square units. Now I'm rechecking my calculations frantically. Lol.

I guess there's no rule against posting your work.

I divided it in half down the middle and made a note to double my answer at the end.

The whole half rectangle has an area of 20. The trapezoid below has an area of (2.5+4)*4/2=13. The triangle above has an area of 3*2.5/2=3.25. 20-13-3.25=3.75. Multiplying that by 2 (because that 3.75 is only the area of the top left half of the shaded portion) gives 7.5.

Edit: Bah! Never mind. Found my mistake. 3*2.5/2=3.75, not 3.25. With that fix I get 6.5.

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zkfcfbzr 4 points 2 years ago

Posting work is encouraged 🙂

You've got it, it's 6.5. I actually posted this problem because I originally found the answer using trig, which seemed a bit too brute-force-y, especially considering the original source for this problem - I wanted to see if others could/would find the simpler solution that I assumed existed. And you did ^_^

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sin_free_for_00_days 4 points 2 years ago

I tried it just using the area of a kite, but ended up with sqrt(2)*sqrt(22.25) and got ~6.67 I made a mistake somewhere I guess.

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zkfcfbzr 6 points 2 years ago

The issue there is it isn't a kite: The two longer sides don't have equal length - they're sqrt(73)/2 and sqrt(61)/2. So it's a decent approximation but not quite exact.

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HexesofVexes 4 points 2 years ago

13/4 was my original guess, when I checked I realised I'd forgotten to multiply by 2.

I used the same method as clara (and my solution would have matched ^_^)

(Spent a lot of time convincing myself they met in the center)

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Grinning 4 points 2 years ago

It is symmetrical across the diagonal so we will only look at the top left quarter, then multiply by 2 when we are done. Because it is 1/4 of the rectangle it has area 1/4 of the original. 1/4 x 8x5 = 10. This leaves two right triangles and the grey area. The top right triangle has a base of (8/2)-1=3 And height of 5/2=2.5 Area = 1/2 x 3 x 2.5 = 3.75 The bottom left triangle has height of (5/2)-1=1.5 And base of 8/2=4 Area = 1/2 x 1.5 x 4 = 3 Top left grey area is the area of the the quarter minus the areas of the two right triangles = 10 - 3.75 - 3 = 3.25 There are two of these gray areas so final answer is 2 x 3.25 = 6.5

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siriusmart 4 points 2 years ago

I love doing things the wrong way, so ::: spoiler spoiler :::

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Revonult 3 points 2 years ago

Also got 6.5. Calculated one section of white space as a sum of a square and two triangles. Then doubled it and subtracted from overall.

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