Solve the problem.
-The perimeter of a triangle is 81 cm. The triangle is isosceles now, but if its base were lengthened by 4 cm and each leg were shortened by 5 cm, it would be equilateral. Find the length of the base
Of the original triangle.
A) 25 cm
B) 20 cm
C) 21 cm
D) 30 cm

Question

Quizplus · Accepted Answer

The original isosceles triangle has two legs of equal length and a base. Let's denote the length of each leg as $L$ and the base as $B$. The perimeter is given as $81$ cm, so we have the equation $2L + B = 81$.When the base is lengthened by $4$ cm and each leg is shortened by $5$ cm to form an equilateral triangle, all sides of the new triangle are equal. This means the new base is $B + 4$ and the new length of each leg (which is now equal to the base in the equilateral triangle) is $L - 5$. Therefore, we have $B + 4 = L - 5$.Since it's an equilateral triangle, all sides are equal, and the perimeter remains $81$ cm. Thus, $3(B + 4) = 81$ or $3L - 15 = 81$. Solving the first equation for $B$, we get $B = L - 9$.Substituting $B = L - 9$ into the perimeter equation of the original triangle, $2L + (L - 9) = 81$, simplifies to $3L = 90$, giving $L = 30$. Substituting $L = 30$ into $B = L - 9$ gives $B = 21$.Therefore, the length of the base of the original triangle is $21$ cm.

Solve the Problem. -The Perimeter of a Triangle Is 81 Cm. the Triangle