the area of an isosceles triangle is 12cm^2 and the length of its equal sides is 5cm, find the base and the height
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If an isosceles triangle has an area of 12 cm² and equal sides are 5 cm, how do I find the length of its base?
Scott Kreidler
Answered 2 years ago
If an isosceles triangle has an area of 12 cm²and equal sides are 5 cm, how do I find the length of its base?
There are actually two solutions to this problem!
We know that A=12bh=12 , and that a vertical line from the apex of the triangle to the base will divide it into two congruent right triangles, each with base 12b , height h , and hypotenuse 5.
However, to make the math a little simpler, I’m going to define a new variable, c=12b , so A=ch=12 , and each right triangle has a height of c .
From there, the Pythagorean theorem tells us:
c²+h²=52
By this point, you might have already figured out what two numbers multiply to make 12, and make a Pythagorean triple with 5, but let’s continue!
Solving the area equation for h , we get h =12c . Substituting that expression into the Pythagorean equation, we get:
c²+(12c)²=25
Multiplying through by c² , we get:
c⁴+144=25c²⟹c⁴-25c²+144=0
We can factor that into (c²−9)(c²−16)=0 , and solve those two expressions to get:
c={±3,±4}
The negative values have no meaning in the context of triangle dimensions, so c={3,4) .
Returning to c=12b , b=2c={6,8} . That is, the possible solutions are:
Base of 6 cm, height of 4cm
Base of 8 cm, height of 3 cm