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

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​

Everleigh · Answer

large sf underline{Given ::}      Altitude of a triangle is 6 cm more than it's base.         Area of the triangle =   sf :180cm^{2}      ‎       large sf underline{To :find ::}      Base and altitude of the triangle.      ‎       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!     ‎       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  .        ‎       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      ‎       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         ‎   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}}}}  ★    ===================     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~!! }     =================‎          Final     Answers    :     Base =  12 cm       Altitude =  1  8     cm ‎      ‎   !! Hope it helps !!

Vivian

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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​