Believe in geometry, not algebra

How can we find the area of the rectangle inside of a right triangle when we don't know the sides of the rectangle?

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#math #algebra #mathbasics

[solidpixel]

another way to do it is to let the height of the rectangle = y and length of rectangle = x. Therefore y/4 = 6/x. Hence xy = 24 since the two white triangles have the same tangent.

[abdo-hanaka--official]

I love how he is so happy after finishing a problem. Always nice to see a man love what he is doing❤️

[m.h.]

Solution:



The two triangles are mathematically 'similar'.
If you assign h as the height of the rectangle and w as the width, than - because of the similarity - we can say, that:
6/w = h/4 |*4w
24 = h * w
And the rectangle area is h * w, so we already have our answer.

[luisfilipe]

I did it by triangle equivalence and cross multiplication. This is smarter

[RB_Universe_TV]

The both 'small' triangles have 3 same angles
So, they are identical

So, x/6 = 4/y
xy = 24

[jubinsoni]

Assuming length of rectangle as x and breadth as y

Then

1/2(6x) + 1/2(4y) + xy = 1/2(4+x)(6+y)
i.e xy = 24

[Itstoearly]

I just made x and y the base and height of the rectangle, and the used the ratio of the 2 legs of the large triangle as equal to the 2 legs of one of the smaller triangles (since there are 3 similar triangles). Therefor 4/y (ratio of the sides of the smaller left triangle) = x+4/y+6 (ratio of the sides of the largest triangle). Simplify it and you get x*y=24

[.sukyar]

Knowing similarity, even though this proof is true aswell, i wanna cry 😢

[cyrusyeung]

I love your "completing the rectangle" method. Here is how I solve it:

Let
x cm be the length of the upper side of the rectangle
y cm be the length of left side of the rectangle

Notice that the small triangles (above and to the left of the rectangle) are similar. (can be proven rigorously by comparing the interior angles)

Hence, x/6 = 4/y, which means
xy = 6 × 4 = 24

area of rectangle = xy = 24 (cm²)

[goatfiddler]

If there had been the need for another triangle, what would the "equality face" have been???? ;)

[experimentingalgorithm]

Solved it using similarity of triangles, btw your solution's cool tho

[chesssofia]

i solved it using similar triangles the two triangles are similar with two sides, 4 and 6/x and 6 and 4x hence area = 6/x * 4x = 24

[ChrisHinton]

Looks like I did it the long way:

(rectangle area) Ra = xy
(triangle area) Ta = 0.5(4 + x)(6 + y)
(top small triangle area) T1a = 0.5*6x = 3x
(bottom small triangle area) T2a = 0.5*4y = 2y
Ra = xy and also Ra = Ta - T1a - T2a
xy = 0.5(4 + x)(6+y) - 3x - 2y
xy = 0.5(24 + 4y + 3x + xy) - 3x - 2y
xy = 12 + 2y + 3x + 0.5xy -3x - 2y
xy = 12 + 0.5xy
0.5xy = 12
xy = 24

[zachansen]

Can someone show me how to solve this with limits? Limit as the bottom left angle approaches zero or ninety? It must be the same no matter what those angles are but I just can't figure it out.

Or just some sort of trig proof that doesn't assume what the angles are as long as they are valid for a right triangle?

[neilgerace]

I saw it as 3-4-5 and 6-8-10 right triangles. So I got the answer as 3 x 8 which are the dimensions of the rectangle. The area is still 24.

[olofmasteryt]

this was the exact problem the teacher gave us in calculus, optimization of the max area of a rectangle inside a triangle