SOLUTION: In an isosceles triangle ABC, side AB is twice as long as AC. The perimeter of the triangle is 200cm. Find the length of AC.

Question 1191970: In an isosceles triangle ABC, side AB is twice as long as AC. The perimeter of the triangle is 200cm. Find the length of AC.

Found 4 solutions by math_tutor2020, ikleyn, MathTherapy, greenestamps:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

x = length of side AC
2x = length of side AB, since it's twice as long as AC

Triangle ABC is isosceles, meaning there are exactly two sides that are the same length.
We've already ruled out AB and AC being the same length.

It's not clear if BC = AC or if BC = AB

Both cases cannot be true simultaneously. Why not? Because that would lead to the false statement that AB = AC, when it should be AB > AC.

If BC = AC, then BC = x and,
AB+BC+AC = perimeter
2x+x+x = 200
4x = 200
x = 200/4
x = 50
So AC = 50

Or if BC = AB, then BC = 2x and,
AB+BC+AC = perimeter
2x+2x+x = 200
5x = 200
x = 200/5
x = 40
meaning that AC = 40

So there are two possibilities:
Either AC = 50 or AC = 40
depending on which two sides are equal of this isosceles triangle.

Currently there isn't enough information to determine which of those values is the correct AC length.

Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
In an isosceles triangle ABC, side AB is twice as long as AC.
The perimeter of the triangle is 200cm. Find the length of AC.
~~~~~~~~~~~~

Since the triangle is isosceles (given) and since side AB is twice as long as AC, it means that AC is the base of the triangle, while AB and BC are the lateral sides. It is only one possible configuration for the triangle to exist. So, if x is the length of the base AC, then the lateral sides AB and BC have the lengths 2x, each. Then for the perimeter we have this equation AC + AB + BC = 200 cm, or x + 2x + 2x = 200, 5x = 200 x = 200/5 = 40. ANSWER. The base AC is 40 cm long. The lateral sides are 2*40 = 80 cm long: they are congruent.

Solved.

Answer by MathTherapy(10551)   (Show Source): You can put this solution on YOUR website!
In an isosceles triangle ABC, side AB is twice as long as AC. The perimeter of the triangle is 200cm. Find the length of AC.

Scenario 1. Let's assume that AC is one of the congruent sides. Then BC is the other congruent side and AB would be the base. Let AC be x. Then BC will also be x, and AB would be 2x. With perimeter being 200 cm, we get: x + x + 2x = 200 4x = 200 x, or Thus, we have the following: AC = 50 cm, BC = 50 cm, and AB = 2(50) = 100 cm. According to the TRIANGLE INEQUALITY, the 3rd side of ANY TRIANGLE, MUST be GREATER than the DIFFERENCE of the other 2 sides, but less than their SUM. In this case, it would be: (50 - 50) < 3rd side < (50 + 50), which results in: 0 < 100 < 100, which is FALSE. Scenario 2. We now make the longer sides congruent Then the base, AC would be the shortest side Let AC be x. Then AB and BC would be 2x, each. With perimeter being 200 cm, we get: x + 2x + 2x = 200 5x = 200 x, or Thus, we have the following: AC = 40 cm, AB = BC = 2(40) = 80 cm. According to the TRIANGLE INEQUALITY, the 3rd side of ANY TRIANGLE, MUST be GREATER than the DIFFERENCE of the other 2 sides, but less than their SUM. In this case, it would be: (80 - 80) < 3rd side < (80 + 80), which results in: 0 < 40 < 160, which is TRUE. Therefore, AC's length is: 40 cm.

Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!

One side is twice the length of the other, so we can call those two side lengths x and 2x.

And the triangle is isosceles, so the length of the third side is either x or 2x.

But the lengths x, x, and 2x do not form a triangle.

So the length of the third side is 2x.

The perimeter is 200cm:

x+2x+2x = 200
5x = 200
x = 40
2x = 80

The triangle side lengths are 40, 80, and 80.

The question asks for the length of AC, which is the shortest side.

ANSWER: AC = 40cm

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