Question

2. In a quadrilateral field ABCD, angleB = 90°, AB = 80 m, BC = 150 m, CD = 120 m and
DA = 250 m. Find the area of the field in hectares. Find the price of the field if
it costs Rs12,000/ hectare.​

Answer 1

Answer:

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Answer 2

$\huge \bold \color{green} \mid \mid{ \underline{ \underline \red{required \: answer}}} \mid \mid</p><p>$

$\bold { {\underline{ \underline \orange{given}}}}</p><p>$

angle B = 90°

AB = 80m

BC = 150m

CD = 120m

DA = 250m

$\bold { {\underline{ \underline \orange{answer}}}}</p><p> \:$

Rs. 18,000

$\bold { {\underline{ \underline \orange{solution}}}}</p><p> \:$

In right angle ∆ ABC

AB² + BC² = AC² ( by Pythagoras theorem )

=> (80)² + (150)² = AC²

=> 6400 + 22500 = AC²

=> 28900 = AC

=> AC = 170m

$ar( \triangle) = \frac{1}{2} \times b \times h \\ \\ \implies ar(ABC) = \frac{1}{2} \times 150 \times 80 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 150 \times 40 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 6000 {m}^{2} ............ar(i)$

now we have to find area of ∆ACD by heron's formula

$ar( \triangle) = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ s = \frac{a + b + c}{2}$

so in triangle ACD

$s = \frac{AC + CD+ DA}{2} \\ \\ s = \frac{170 + 120 + 250}{2} \\ \\ s = \frac{540}{2} \\ \\ s = 270$

then area of triangle ACD,

$ar(acd) = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = \sqrt{270(270 - 250)(270 - 120)(270 - 170)} \\ \\ = \sqrt{270 \times 20 \times 150 \times 100} \\ \\ = \sqrt{27 \times 10 \times 2 \times 10 \times 15 \times 10 \times 10 \times 10} \\ \\ = 100 \sqrt{27 \times 2 \times15 \times 10 } \\ \\ = 100 \sqrt{3 \times 3 \times 3 \times 2 \times 3 \times 5 \times 2 \times 5} \\ \\ = 100 \times 3 \times 3 \times 2 \times 5 \\ \\ = 9000 {m}^{2} ............ar(ii)$

now area of quadrilateral ABCD is;

ar(ABCD) = ar(ABC) + ar(ACD)

$\implies \: ar(abcd) = 6000 + 9000 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 15000 {m}^{2}$

so area of quadrilateral field is 15000m²

we know that..

1 hectare = square metre / 10000

$\bold {\:so \: 15000 {m}^{2} = 1.5hectare}$

so area of field is 1.5 Hectare

cost = 12,000 / hectare

then price of field is;

$\bold{1.5 \times cost \: of \: field} \\ \\ \bold{\implies \: price \: = 1.5 \times 12000} \\ \\ \bold{\implies \: price \: of \: field \: = 18000 \: rs }$

hence, price of the field is Rs. 18000