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11Alexandr11 [23.1K]
3 years ago
11

In triangle MNO shown below, segment NP is an altitude:

Mathematics
1 answer:
Effectus [21]3 years ago
5 0

Answer:  C. Definition of an Altitude

Step-by-step explanation:

Given: In triangle MNO shown below, segment NP is an altitude from the right angle.

Let ∠MNP=x

Then ∠PNO=90°-x

Therefore in triangle MNO,

∠MPN=∠NPO =90°  [by definition of Altitude]

[Definition of altitude : A line which passes through a vertex of a triangle, and joins the opposite side forming right angles. ]

Now using angle sum property in ΔMNP

∠MNP+∠MPN+∠PMN=180°

⇒x+90°+∠PMN=180°

⇒∠PMN=180°-90°-x

⇒∠PMN=90°-x

Now, in ΔMNO and ΔPNO

∠PMN=∠PNO=90°-x

and ∠MPN=∠NPO =90°  [by definition of altitude]

Therefore by AA similarity postulate, we have

ΔMNO ≈ ΔPNO

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Answer:

I = 91.125

Step-by-step explanation:

Given that:

I = \int \int_E \int zdV where E is bounded by the cylinder y^2 + z^2 = 81 and the planes x = 0 , y = 9x and z = 0 in the first octant.

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since curve z = 0 and y^2 + z^2 = 81

∴ z^2 = 81 - y^2

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Thus, z lies between 0 to \sqrt{81 - y^2}

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x =\dfrac{y}{9}

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Given curve x = 0 , x =\dfrac{y}{9} and z = 0, y^2 + z^2 = 81

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I = \int^9_{y=0}  \begin {bmatrix}  \dfrac{{81(\dfrac{y}{9}) -(\dfrac{y}{9})y^2} }{2}-0 \end {bmatrix}     \ dy

I = \int^9_{y=0}  \begin {bmatrix}  \dfrac{{81 \  y -y^3} }{18} \end {bmatrix}     \ dy

I = \dfrac{1}{18} \int^9_{y=0}  \begin {bmatrix}  {81 \  y -y^3}  \end {bmatrix}     \ dy

I = \dfrac{1}{18}  \begin {bmatrix}  {81 \ \dfrac{y^2}{2} - \dfrac{y^4}{4}}  \end {bmatrix}^9_0

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I = 91.125

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Step-by-step explanation:

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=============================================================

Explanation:

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----------------

The slightly longer method involves letting x be the measure of the missing interior angle of the triangle.

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