Calculate the distance BD | Important Geometry and Algebra skills explained

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Calculate the distance AB | Important Geometry and Algebra skills explained

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[yuusufliibaan]

Thank you teacher for teaching lesson is the best way keep going dear teacher i say always happy day

[اممدنحمظ]

تمرين جميل جيد. رسم واضح. شرح واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين .

[tompeled]

*Step 1: Find the area of triangle ABC via Heron's formula*
A=√[s(s-a)(s-b)(s-c)]
A=√[10(10-8)(10-7)(10-5)]
A=√[10(2)(3)(5)]
A=√300
A=10√3

*Step 2: Find BP via plugging the area into the base×height formula*
bh/2=A
8(BP)/2=10√3
8(BP)=20√3
BP=20√(3)/8
BP=5√(3)/2

*Step 3: Find PC via the Pythagorean theorem*
a²+b²=c²
a²+[5√(3)/2]²=5²
a²+[5√(3)]²/2²=25
a²+5²(√3)²/4=25
a²+25•3/4=25
a²+75/4=25
a²=25-75/4
a²=100/4-75/4
a²=25/4
a=√(25)/√4
a=5/2
PC=5/2

*Step 4: Find PA via subtraction*
AC-PC=PA
8-5/2=PA
16/2-5/2=PA
11/2=PA

*Step 5: Find PD via intersecting chords theorem*
11/2(5/2)=x[5√(3)/2]
55/4=x[5√(3)/2]
55/4÷[5√(3)/2]=x
55(2)/4[5√3]=x
110/20√3=x

11√(3)/6=x
PD=11√(3)/6

*Step 6: Find BD via addition*
BP+PD=BP
11√(3)/6+5√(3)/2=BD
11√(3)/6+15√(3)/6=BD
26√(3)/6=BD
13√(3)/3=BD

*Final Answer*
BD=13√(3)/3
BD≈7.51

[nakamakai]

Brilliant - you just keep them coming, day after day - Nice - We love it

[AnonimityAssured]

Some very useful techniques here. I'm especially pleased to see this problem, as I'm working on what could be called Pythagorean kites, which are quadrilaterals consisting of four Pythagorean triangles, arranged such that their right angles meet at a point, and all line segments are of whole-number length. Asymmetrical Pythagorean kites are particularly interesting, as their corners lie on a circle, just as in this problem.

[anatoliy]

Very nice task. Inordinary application of the triangulation method. Thank you so much for your lesson, sir. All the best to you, Mr PreMath.

[DDX]

Very nice I have also solved by myself.
But but but all credit goes to you sir....

[hadigayar]

i had solved it in different way and got the same result
thank u sir .that was amazing question

[murdock]

Simply clever, many thanks, Sir! Every lesson is a surprise - this channel is extraordinary great! 🙂

[mathrapper]

Good question applying Pythagorean theorem and chords theorem!

[rudychan]

Instantly, i recognize Triangle: 5-7-8 is special. ∆
You Should use Cosinus' formula. (≤ BCA = ≤ C)
cos C = 5"+8"-7" / 2•5•8 = 40/80 = 1/2 ➡ C = 60°!
...
But, Amazing Formula: a•b = c•d, I forget how to proof it/ I forget this formula^ 🤓

[alexundre]

Good Morning Master
Thank you so much Sir 🇧🇷

[jackhusky]

I used Herons Law to calculate Area of ABC then worked out BP as Area ABC and base 8 now known.

[sadafkhanam]

Nice question 👍
Thanks sir for this question 😊

[manojitmaity]

Sir please let me know from which book you collect these problems, also add its price...

[AHMED_-..]

What program do you use to draw the shapes...and write the solution, please