Question 442432: area of an isosceles triangle is 60sq. cm. the same two side of is 13 cm, what is the other one leangth ??
Answer by solver91311(24713)   (Show Source):
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Construct triangle ABC with the vertex B common to the two congruent sides upward and the unknown side as the base. Construct a perpendicular to the base through the vertex at the top of the triangle. Label the point of intersection of the perpendicular and the base with D.
Your picture should look like one of the following:
Let  represent the measure of segement AD. Let  represent segment BD. Since we know from a common geometric proof that the perpendicular from the apex to the base of an isosceles triangle bisects the base, the apex angle, and the triangle itself. Therefore the area of triangle ABD is equal to the area of triangle BCD and the area of each is one-half of the area of the entire triangle ABC.
From the formula for the area of a triangle, we have:
Hence,
And further,
From which we can derive:
From the fact that segment BD is perpendicular to segment AD, we can deduce that triangle ABD is a right triangle, for which we are given the measure of the hypotenuse is 13 and we have identified the measure of the legs as and . Using Pythagoras:
And using what we learned earlier about the relationship between and , we can write:
Multiply both sides by  :
Let  , substitute, and put the resulting quadratic into standard form:
Factor:
(u\ -\ 25)\ =\ 0)
Then
or
Discarding the negative roots because we are attempting to calculate length, we have  or  .
Having previously determined that segment BD bisects segment AC, and by construction, AD plus DC is equal to AC, we can deduce that AC is equal to two times AD. Therefore the base measurement that we seek as an answer to this problem is two times the value of
or
John
My calculator said it, I believe it, that settles it
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