Calculate the height h | Important Geometry and Algebra skills explained

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Calculate the height h | Important Geometry and Algebra skills explained

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[bigm]

Thanks for another stepwise explanation!❤🥂

[misterenter-izrz]

AD is 25, by pythagorean theorem, so DB is 25 in an isosceles triangle, then CB is 25+7=32, and thus AB is 40, by pythagorean theorem again, now EB is one half of AB, is 40/2=20, therefore h=15, also applying pythagorean theorem. 😃

[muttleycrew]

ABC is similar to DBE so once you know that ABC is in the proportions of a 3, 4, 5 triangle (it's 8x3, 8x4, 8x5) you know that DBE must also be in the proportions of a 3, 4, 5 triangle. You know that the hypotenuse is 5x5 = 25 so that leaves the other sides being 20 and 15. From similarity h must be the shortest side so that's h = 15.

[solimana-soli]

Setting AE=EB=x and DB=y, I constructed two equations, the first by applying the Pythagorean theorem to the triangle ABC
(24)^2+(7+y)^2=(2x)^2
I found the second equation by applying the proportions to the two similar triangles ABC and EBD
2x:y=(y+7):x hence 2x^2=y^2+7y
whose solutions are x=20 and y=25
Finally, to find h it is enough to apply the Pythagorean theorem again to the triangle EBD, the sides EB and DB being known

[georgecaplin]

I solved it, (probably the same way most others did), by connecting A and D, making that the hypotenuse of the triangle ACD, solving for the length AD, (which is also DB), which gives us BC, from which you can solve for AB, half of AB is BE, then you have all the values to “plug in to” the Pythagorean theorem.

[BKNeifert]

This is how math should be taught. Geometry is a lot like poetry, when you're watching someone solve it. If I ever taught an English Class, I'd pull up one of these types of videos, and teach them to apply the same skills in this, to their poems. Nobody thinks like that, though, but that's also why we have a lot of people who can't think. They always want words to have ambiguity, but a good reader will follow the logic of the poem's words on the page, the same way they'd follow this guy's presentation. Like, poetry is exactly like this video. You're being walked step by step to the answer. And each verse breaks the logic down into simpler bits. Good poetry, anyway is like that.

[sailbyzantium]

Excellent! Very memorable. I need to do a better job in remembering as you show here that one can actually MANIPULATE figures and shapes to solve them, rather than being just stuck with what is given to try to deductively figure it out. Thank you, PreMath!

[spiderjump]

Find AD by Pythagorean theorem, AD = 25.
Triangles ADE and BDE are congruent. Hence DB =25.
BC =32.
AB can be found by Pythagorean theorem, AB= 40.
EB=40/2=20
DE can be found by Pythagorean theorem, DE= 15

[AnonimityAssured]

An in-the-head quickie if one recognizes two familiar Pythagorean triples.

Spoiler alert.

An auxiliary line segment can be imagined linking points A and D.
We can immediately recognize a 7-24-25 triangle in the top left, so AD = 25 units.
Triangles ADE and BDE are congruent, so DB must also be 25 units.
CB = 7 + 25 = 32 units.
24 is three-quarters of 32, so ABC is a scaled 3-4-5 triangle.
Therefore, AED is also a scaled 3-4-5 triangle, so:
h = 25 (3 / 5) = 15 units.

Edited to correct a small typo.

[HappyFamilyOnline]

Another great video👍
Thanks for sharing😊😊

[latten]

Thanks, it’s intersting. I used AREA to find out h value. Triangle ABD=300, AB=40, 40xhx1/2=300, 20h=300, h=15
Just a comment from Tokyo, Japan.

[alster]

Super easy. After seeing the Similarity Theorem, the problem became a piece of cake and solved it off the bat before I can finish the video.

[liliyakaloyanova]

AD=DB, because triangles ADE and EBD are equal. AD^2=24^2+7^2=576+49=625 AD=DB=25
so CB=32. In triangle ABC AB^2= 24^2+32^2= 576+1024=1600 AB=40, so DB=40/2=20 . In triangle DBE h^2= 25^2- 20^2= 625 - 400 = 225 h=V225=15

[NoraBhsas]

Me having an English degree and not having anything to do with Geometry actually listening and enjoying this explanation

[kevi]

Tan BDE = h/20 =24/32 . So h =3/4 x 20 = 15 . Since triangle ABC is similar to triangle BDE !

[rickhart]

I did the last step without pythagorean theorem:
There are 2 ways to calculate the area of the triangle.
(24/2*32) = (40/2*h) + (24/2*7)
20*h = 12(32-7)
20*h = 12*25
20*h = 300
h =15

[calspace]

I am terrible in geometry at knowing where to draw additional line. I used pure algebra.

Let us call the length of DB = x and the length of EB = y (so AB = 2y).

We have two right triangles, ABC and DBE. Because B is present in both of them, they are similar by AAA. The lengths of the sides of these two triangles (SL;LL;H) are 24;7+x;2y and h;y;x. Therefore, comparing the ratio of the long legs and hypotenuses, we get y:7+x=x:2y. When we multiply this, we get 2y^2 = (7+x)*x = x^2 + 7x.

By the Pythagorean theorem, 24^2 + (7 + x)^2 = (2y)^2
576 + 49 + 14x + x^2 = 4y^2

If we multiply the first equation by 2 we get 4y^2 = 2x^2 + 14x.

Therefore, 625 + 14x + x^2 = 2x^2 + 14x.

Substracting x^2 + 14x from both sides, we get 625 = x^2, or x = 25.

So now we know that the sides of ABC are 24; 7 + 25 = 32; 2y.

Using the Pythagorean formula we get 576 + 1024 = 4y^2
1600 = 4y^2
400 = y^2
20 = y

Now we can either use the Pythagorean formula again to get x^2 - y^2 = x^2
25^2 - 20^2 = h^2
625 - 400 = h^2
225 = h^2
15 = h


Or we can use the ratios of sides, either 24:h = 7+x:y
24y = (7 + x)h
24 * 20 = 32h
480 = 32h
480/32 = h
15 = h

or 24:h = 2y:x
24x = 2y * h
24 * 25 = 2 * 20 * h
600 = 40h
600/40 = h
15 = h

[rudychan]

More than one hour, I figured it out, h = 15! 😓
I was connecting line CE = AE = BE = it's a Radius!* (Fail)
Then, I connect AD, voila!
Much-much Easier!
It is Triple Phytagoras!!!
Great Maths 🙂 👍

[juanpabloalvarado]

Thanks for the explanation, needs for me to see 100 times until I try by myself but I going to get it, Thanks again

[victorfildshtein]

let's assume that AE=BE=x
△DEB and △ACB triangles are similar
We make proportions:
EB/BC = CE/AB ⇒ x/BC = (BC - 7)/2x ⇒
2x² = BC² - 7BC (1)
According to the Pythagorean theorem
BC² = AB² - AC² = 4x² - 24² (2)
Substitute (2) into (1)
2x² = 4x² - 24² -7√(4x² - 24²)
7√(4x² - 24²) = 2x² - 24²
We get the biquadratic equation
4x⁴ - 2500x² + 360000 = 0
x₁² = 400; x₂² = 225
x₁ = 20; x₂ = 15
1)Consider the first option: x = 20
AC = 40; BC = √(40² - 24²) = 32; EC = 25
h = √(25² - 20²) = 15
2) Consider the second option: x = 15
AC = 30; BC = √(30² - 24²) = 18; EC = 11
EC < EB the hypotenuse is smaller than the leg. This is impossible.
Answer: h = 15