Question

the altitudes of a triangle is 6cm more than its base if it's areas is180 cm 2 . find the base and altitudes of the triangle​

Answer 1

$\large\sf\underline{Given\::}$

• Altitude of a triangle is 6 cm more than it's base.

• Area of the triangle = $\sf\:180cm^{2}$

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$\large\sf\underline{To\:find\::}$

• Base and altitude of the triangle.

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$\large\sf\underline{Concept\::}$

Here in the question we are given the relation between altitude and base and the area of the triangle. We are asked to calculate the exact values of base and altitude. For doing so we need to first form an expression for base from the given relation. Doing so we could equate the formula for area and the given value of area and solving that we could get the final answers. Let's begin!

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$\large\sf\underline{Formula\:to\:be\:used\::}$

• Area of the triangle = $\sf\:\frac{1}{2} \times b \times h$

where, b stands for breadth and h stands for height.

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$\large\sf\underline{Assumption\::}$

Let the base of the triangle be b.

Therefore according to the question :

Altitude of the triangle is 6 cm more than it's base.

• i.e., ( b + 6 ) cm

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$\large\sf\underline{Solution\::}$

Now equating the formula of area and the given value of area :

$\sf\:\frac{1}{2} \times b \times h=180cm^{2}$

• Substituting th values of base and altitude that we have assumed

$\sf\implies\:\frac{1}{2} \times b \times (b+6)=180$

• Multiplying the terms

$\sf\implies\:\frac{1}{2} \times b^{2}+6b=180$

$\sf\implies\:\frac{b^{2}+6b}{2}=180$

• Cross multiplying

$\sf\implies\:b^{2}+6b=180 \times 2$

$\sf\implies\:b^{2}+6b=360$

$\sf\implies\:b^{2}+6b-360=0$

• Factorising by middle term breaking

$\sf\implies\:b^{2}+(18-12)b-360=0$

$\sf\implies\:b^{2}+18b-12b-360=0$

$\sf\implies\:b(b+18)-12(b+18)=0$

$\sf\implies\:(b+18)(b-12) = 0$

Case 1 -

$\sf\:b+18=0$

$\bf\to\:b=-18$

Case 2 -

$\sf\:b-12=0$

$\bf\to\:b=12$

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Since the measure of a base ( b ) can't be negative the value of base is $\small{\underline{\boxed{\mathrm\red{12\:cm}}}}$ ★ .

Now let's calculate the value of altitude oR height by substituting the value of b :

Assumed value of altitude : ( b + 6 ) cm

So Altitude : ( 12 + 6 ) cm = $\small{\underline{\boxed{\mathrm\red{18\:cm}}}}$ ★

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Verifying :

For being sure whether our answers are correct we need to substitute the values of b and h in the formula and equate it with the given value of area. Doing so if we get LHS = RHS our answers would be correct.

$\sf\:\frac{1}{2} \times b \times h=180cm^{2}$

• Substituting the values of b and h

$\sf\rightarrow\:\frac{1}{2} \times 12 \times 18=180$

• Multiplying the numbers

$\sf\rightarrow\:\frac{360}{2}=180$

• Reducing the fraction to lower terms

$\sf\rightarrow\:\cancel{\frac{360}{2}}=180$

$\sf\rightarrow\:180=180$

$\bf\rightarrow\:LHS=RHS$

$\small\fbox\blue{Hence~Verified~!! }$

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Final Answers :

• Base = 12 cm

• Altitude = 18 cm‎

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!! Hope it helps !!

Answer 2

Answer:

answer above

hope it helps