Answer:
Step-by-step explanation:
Xét tam giác AHC và tam giác BAC có
góc AHC = góc BAC =90°
Góc C chung
Suy ra tam giác AHC đồng dạng với tam giác BAC
Suy ra AH×BC=AB×AC
Suy ra h.a=b.c
Suy ra h^2.a^2=b^2.c^2
Suy ra h^2.(b^2+ c^2 )=b^2.c^2
( áp dụng định lí pytago vào tam giác ABC)
Suy ra 1/h^2=c^2+ b^2/b^2.c^2
Suy ra 1/h^2=1/b^2+1/c^2
Step-by-step explanation:
[tex] \bf \underline{Given \:Question -} \\ [/tex]
In ∆ ABC, ∠C = 90°, CD ⏊ AB, CD =h, BC = a, CA = b, AB = c, Prove that
[tex] \sf \: \dfrac{1}{ {h}^{2} } = \dfrac{1}{ {a}^{2} } + \dfrac{1}{ {b}^{2} } [/tex]
[tex] \red{\large\underline{\sf{Solution-}}}[/tex]
Given that,
In ∆ABC, ∠C = 90° and BC = a, CA = b, AB = c
So, Area of ∆ ABC is given by
[tex]\rm :\longmapsto\: Area_{∆ABC} = \dfrac{1}{2} \times ab - - - (1)[/tex]
Also,
In ∆ ABC, CD ⏊ AB
So, Area of ∆ABC is given by
[tex]\rm :\longmapsto\: Area_{∆ABC} = \dfrac{1}{2} \times ch - - - (2)[/tex]
From equation (1) and (2), we get
[tex]\rm :\longmapsto\:\dfrac{1}{2}ab = \dfrac{1}{2}ch[/tex]
[tex]\rm :\longmapsto\:ab = ch[/tex]
[tex]\rm\implies \:\boxed{\tt{ c = \frac{ab}{h}}} - - - (3)[/tex]
Again, In right ∆ ABC
Using Pythagoras Theorem, we have
[tex]\rm :\longmapsto\: {a}^{2} + {b}^{2} = {c}^{2} [/tex]
On substituting the value of c, from equation (3), we get
[tex]\rm :\longmapsto\: {a}^{2} + {b}^{2} = {\bigg[\dfrac{ab}{h} \bigg]}^{2} [/tex]
[tex]\rm :\longmapsto\: {a}^{2} + {b}^{2} = \dfrac{ {a}^{2} {b}^{2} }{ {h}^{2} } [/tex]
can be rewritten as
[tex]\rm :\longmapsto\:\dfrac{ {a}^{2} + {b}^{2} }{ {a}^{2} {b}^{2} } = \dfrac{1}{ {h}^{2} } [/tex]
[tex]\rm :\longmapsto\:\dfrac{ {a}^{2}}{ {a}^{2} {b}^{2} } + \dfrac{ {b}^{2}}{ {a}^{2} {b}^{2} }= \dfrac{1}{ {h}^{2} } [/tex]
[tex]\rm\implies \:\boxed{\tt{ \frac{1}{ {a}^{2} } + \frac{1}{ {b}^{2} } = \frac{1}{ {h}^{2} }}}[/tex]
Hence, Proved.
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1. Pythagoras Theorem :-
This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.
2. Converse of Pythagoras Theorem :-
This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.
3. Area Ratio Theorem :-
This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.
4. Basic Proportionality Theorem:
If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.