Question

The angle of elevation of the top of a tower
OP from a point a on the ground is 45°.
If
height of the tower is 6 cm then find
the area of the triangle OPQ.
​

Answer 1

$\: \: \: \: \: \: \: \: \bullet\bf\: \: \: {PO = height}$

$\: \: \: \: \: \: \: \: \bullet\bf\: \: \: {OQ = base}$

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Given

• The angle of elevation of the top of a tower
• OP from a point a on the ground is 45°.
• Height of the tower is 6cm.

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To find

• Area of the triangle OPQ.

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Solution

• We have the angle of elevation = 45°.

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$\: \: \: \: \: \: \: \: \: \:\underline{\sf{\red{We\: know\: that}}}$

• $\sf{tan\theta = \dfrac{Perpendicular}{Base}}$

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Therefore,

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$\tt:\implies\: \: \: \: \: \: \: \: {tan45^{\circ} = \dfrac{PO}{OQ}}$

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$\: \: \: \: \: \: \: \: \: \:\underline{\sf{\red{tan45^{\circ} = 1}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {1 = \dfrac{6}{OQ}}$

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$\bf:\implies\: \: \: \: \: \: \: \: {OQ = 6}$

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• We get, OQ = 6cm

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Now,

$\large{\boxed{\boxed{\sf{Area_{Triangle} = \dfrac{1}{2} \times b \times h}}}}$

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Here,

• b = base = 6cm
• h = height = 6cm

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$\: \: \: \: \: \: \: \: \: \:\underline{\sf{\red{Putting\: the\: values}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = \dfrac{1}{\cancel{2}} \times \cancel{6} \times 6}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = 3 \times 6}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = 18}$

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Hence,

• Area of the triangle is 18cm².

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Answer 2

Question :

The angle of elevation of top of a tower OP from a point on the ground is 45° .If height of tower is 6cm then find the area of the triangle OPQ.

Given :

• Angle of elevation = 45°
• OP = (Height) = 6cm

To find :

• Area of triangle

Solution :

To find the area of triangle we need to first find the base of the triangle.

So,

$\boxed {\tt tan\theta = \dfrac{Perpendicular}{Base} }$

$\boxed {\underline{\bf tan45° = 1 }}$

$:\implies \tt {tan45° = \dfrac{OP}{OQ}}$

$: \implies \tt { 1 = \dfrac {6}{OQ}}$

$: \implies \tt {OQ= 6cm }$

$\boxed {\bf Area \: of \: triangle = \dfrac {1}{2} \times base \times height }$

$: \implies \tt { Area \: = \dfrac {1}{2} \times 6 \times 6 }$

$: \implies \tt {Area = 18 cm^2 }$