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ANTONII [103]
3 years ago
7

Prove the Cyclic Quadrilateral Conjecture

Mathematics
1 answer:
Rufina [12.5K]3 years ago
4 0

Given ABCD is a cyclic quadrilateral.

To prove that ∠A + ∠C = 180° and ∠B + ∠D = 180°

Construction: Join AC and BD.

Proof:

<em>Angles in the same segment are equal.</em>

∠ACD = ∠ABD – – – – (1)

∠DAC = ∠DBC – – – – (2)

Add (1) and (2), we get

∠ACD + ∠DAC = ∠ABD + ∠DBC

⇒ ∠ACD + ∠DAC = ∠ABC  (Since ∠ABD + ∠DBC = ∠ABC)

Add ∠ADC on both sides.

⇒ ∠ACD + ∠DAC + ∠ADC = ∠ABC + ∠ADC

We know that <em>sum of the angles of a triangle is 180°.</em>

In ΔABC, ∠ACD + ∠DAC + ∠ADC = 180°

Now, 180° = ∠ABC + ∠ADC

180° = ∠B + ∠C – – – – (3)

<em>Sum of all the angles of a quadrilateral = 360°</em>

∠A + ∠B + ∠C + ∠D = 360°

∠A + ∠C + 180° = 360° (using (3))

∠A + ∠C = 360° – 180°

∠A + ∠C = 180°  – – – – (4)

Equation (3) and (4) shows that,

Opposite angles of a cyclic quadrilateral are supplementary.

Hence proved.

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Sanya has a piece of land which is in the shape of a rhombus. She wants her one daughter and one son to work on the land and pro
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{\large{\textsf{\textbf{\underline{\underline{Given :}}}}}}

★ Sanya has a piece of land which is in the shape of a rhombus.

★ She wants her one daughter and one son to work on the land and produce different crops, for which she divides the land in two equal parts.

★ Perimeter of land = 400 m.

★ One of the diagonal = 160 m.

{\large{\textsf{\textbf{\underline{\underline{To \: Find :}}}}}}

★ Area each of them [son and daughter] will get.

{\large{\textsf{\textbf{\underline{\underline{Solution :}}}}}}

Let, ABCD be the rhombus shaped field and each side of the field be x

[ All sides of the rhombus are equal, therefore we will let the each side of the field be x ]

Now,

• Perimeter = 400m

\longrightarrow  \tt AB+BC+CD+AD=400m

\longrightarrow  \tt x + x + x + x=400

\longrightarrow  \tt 4x=400

\longrightarrow  \tt  \: x =  \dfrac{400}{4}

\longrightarrow  \tt x= \red{100m}

\therefore Each side of the field = <u>100m</u><u>.</u>

Now, we have to find the area each [son and daughter] will get.

So, For \triangle ABD,

Here,

• a = 100 [AB]

• b = 100 [AD]

• c = 160 [BD]

\therefore \tt Simi \:  perimeter \:  [S] =  \boxed{ \sf \dfrac{a + b + c}{2} }

\longrightarrow \tt S = \dfrac{100 + 100 + 160}{2}

\longrightarrow \tt S =  \cancel{ \dfrac{360}{2}}

\longrightarrow \tt S = 180m

Using <u>herons formula</u><u>,</u>

\star \tt Area  \: of  \: \triangle = \boxed{\bf{{ \sqrt{s(s - a)(s - b)(s - c) } }}} \star

where

• s is the simi perimeter = 180m

• a, b and c are sides of the triangle which are 100m, 100m and 160m respectively.

<u>Putt</u><u>ing</u><u> the</u><u> values</u><u>,</u>

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} =  \tt \sqrt{180(180 - 100)(180 - 100)(180 - 160) }

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} =  \tt \sqrt{180(80)(80)(20) }

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} =  \tt \sqrt{180 \times 80 \times 80 \times 20 }

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} =  \tt \sqrt{9 \times 20 \times 20 \times 80 \times 80}

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} =  \tt \sqrt{ {3}^{2} \times  {20}^{2}  \times  {80}^{2}  }

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} =  3 \times 20 \times 80

\longrightarrow \tt  Area_{ ( \triangle \:  ABD)} = \red{   4800  \: {m}^{2} }

Thus, area of \triangle ABD = <u>4800 m²</u>

As both the triangles have same sides

So,

Area of \triangle BCD = 4800 m²

<u>Therefore, area each of them [son and daughter] will get = 4800 m²</u>

{\large{\textsf{\textbf{\underline{\underline{Note :}}}}}}

★ Figure in attachment.

{\underline{\rule{290pt}{2pt}}}

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