To find the dimensions of the isosceles triangle given that the perimeter is 50 centimeters and the ratio of the two equal sides to the third side is [tex]\(3:4\)[/tex], we will proceed step by step.
1. Define Variables:
Let the length of the two equal sides be [tex]\(a\)[/tex].
Let the length of the third side (base) be [tex]\(b\)[/tex].
2. Use the Given Ratio:
According to the ratio [tex]\(3:4\)[/tex], we can express the lengths of the sides in terms of a common factor [tex]\(x\)[/tex]:
[tex]\[
a = 3x
\][/tex]
[tex]\[
b = 4x
\][/tex]
3. Express the Perimeter:
The perimeter of the triangle is the sum of all its sides. Therefore,
[tex]\[
2a + b = 50
\][/tex]
Substitute the expressions [tex]\(a = 3x\)[/tex] and [tex]\(b = 4x\)[/tex] into the perimeter equation:
[tex]\[
2(3x) + 4x = 50
\][/tex]
4. Simplify the Equation:
Combine like terms:
[tex]\[
6x + 4x = 50
\][/tex]
[tex]\[
10x = 50
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Divide both sides by 10:
[tex]\[
x = \frac{50}{10}
\][/tex]
[tex]\[
x = 5
\][/tex]
6. Calculate the Lengths of the Sides:
Substitute [tex]\(x = 5\)[/tex] back into the expressions for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a = 3x = 3(5) = 15 \, \text{cm}
\][/tex]
[tex]\[
b = 4x = 4(5) = 20 \, \text{cm}
\][/tex]
7. Verify the Perimeter:
To ensure the accuracy of our solution, check the perimeter:
[tex]\[
2a + b = 2(15) + 20
\][/tex]
[tex]\[
30 + 20 = 50 \, \text{cm}
\][/tex]
Therefore, the dimensions of the isosceles triangle are:
- The two equal sides are each [tex]\(15\)[/tex] cm.
- The third side is [tex]\(20\)[/tex] cm.
