Answer :
Certainly! Let's break down the problem step by step to find the lengths of the sides, the height, and the area of the isosceles triangle.
### Step 1: Understand the problem
The perimeter of the isosceles triangle is 48 cm, and the base (let's denote it as [tex]\( b \)[/tex]) is [tex]\( \frac{3}{2} \)[/tex] times the length of one of the equal sides (let's denote each of these equal sides as [tex]\( a \)[/tex]).
### Step 2: Express the perimeter
Given that the triangle is isosceles, the perimeter is the sum of the lengths of all its sides:
[tex]\[ 2a + b = 48 \][/tex]
### Step 3: Express the base in terms of the equal sides
We know from the problem that:
[tex]\[ b = \frac{3}{2}a \][/tex]
### Step 4: Substitute the base length into the perimeter equation
By substituting [tex]\( b \)[/tex] into the perimeter equation, we get:
[tex]\[ 2a + \frac{3}{2}a = 48 \][/tex]
### Step 5: Solve for [tex]\( a \)[/tex]
Combining the terms involving [tex]\( a \)[/tex]:
[tex]\[ \left(2 + \frac{3}{2}\right)a = 48 \][/tex]
[tex]\[ \frac{4}{2}a + \frac{3}{2}a = 48 \][/tex]
[tex]\[ \frac{7}{2}a = 48 \][/tex]
To isolate [tex]\( a \)[/tex], multiply both sides by [tex]\( \frac{2}{7} \)[/tex]:
[tex]\[ a = \frac{48 \times 2}{7} \][/tex]
[tex]\[ a = \frac{96}{7} \][/tex]
[tex]\[ a \approx 13.714285714285714 \][/tex]
### Step 6: Find the length of the base [tex]\( b \)[/tex]
Using [tex]\( b = \frac{3}{2}a \)[/tex]:
[tex]\[ b = \frac{3}{2} \times 13.714285714285714 \][/tex]
[tex]\[ b \approx 20.57142857142857 \][/tex]
### Step 7: Find the height of the triangle
To find the height [tex]\( h \)[/tex] of the triangle, we use the right triangle formed by drawing a perpendicular from the top vertex to the base. This splits the base into two equal parts of length [tex]\( \frac{b}{2} \)[/tex]:
[tex]\[ \frac{b}{2} = \frac{20.57142857142857}{2} \][/tex]
[tex]\[ \frac{b}{2} \approx 10.285714285714285 \][/tex]
Using the Pythagorean theorem in one of these right triangles:
[tex]\[ a^2 = \left(\frac{b}{2}\right)^2 + h^2 \][/tex]
Solving for [tex]\( h \)[/tex]:
[tex]\[ h^2 = a^2 - \left(\frac{b}{2}\right)^2 \][/tex]
[tex]\[ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} \][/tex]
[tex]\[ h = \sqrt{13.714285714285714^2 - 10.285714285714285^2} \][/tex]
[tex]\[ h \approx 9.071147352221454 \][/tex]
### Step 8: Find the area of the triangle
The area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
[tex]\[ A = \frac{1}{2} \times b \times h \][/tex]
[tex]\[ A = \frac{1}{2} \times 20.57142857142857 \times 9.071147352221454 \][/tex]
[tex]\[ A \approx 93.30322990856351 \][/tex]
### Summary of Results
- Length of each of the equal sides [tex]\( a \approx 13.714285714285714 \)[/tex] cm
- Length of the base [tex]\( b \approx 20.57142857142857 \)[/tex] cm
- Height of the triangle [tex]\( h \approx 9.071147352221454 \)[/tex] cm
- Area of the triangle [tex]\( A \approx 93.30322990856351 \)[/tex] cm²
### Step 1: Understand the problem
The perimeter of the isosceles triangle is 48 cm, and the base (let's denote it as [tex]\( b \)[/tex]) is [tex]\( \frac{3}{2} \)[/tex] times the length of one of the equal sides (let's denote each of these equal sides as [tex]\( a \)[/tex]).
### Step 2: Express the perimeter
Given that the triangle is isosceles, the perimeter is the sum of the lengths of all its sides:
[tex]\[ 2a + b = 48 \][/tex]
### Step 3: Express the base in terms of the equal sides
We know from the problem that:
[tex]\[ b = \frac{3}{2}a \][/tex]
### Step 4: Substitute the base length into the perimeter equation
By substituting [tex]\( b \)[/tex] into the perimeter equation, we get:
[tex]\[ 2a + \frac{3}{2}a = 48 \][/tex]
### Step 5: Solve for [tex]\( a \)[/tex]
Combining the terms involving [tex]\( a \)[/tex]:
[tex]\[ \left(2 + \frac{3}{2}\right)a = 48 \][/tex]
[tex]\[ \frac{4}{2}a + \frac{3}{2}a = 48 \][/tex]
[tex]\[ \frac{7}{2}a = 48 \][/tex]
To isolate [tex]\( a \)[/tex], multiply both sides by [tex]\( \frac{2}{7} \)[/tex]:
[tex]\[ a = \frac{48 \times 2}{7} \][/tex]
[tex]\[ a = \frac{96}{7} \][/tex]
[tex]\[ a \approx 13.714285714285714 \][/tex]
### Step 6: Find the length of the base [tex]\( b \)[/tex]
Using [tex]\( b = \frac{3}{2}a \)[/tex]:
[tex]\[ b = \frac{3}{2} \times 13.714285714285714 \][/tex]
[tex]\[ b \approx 20.57142857142857 \][/tex]
### Step 7: Find the height of the triangle
To find the height [tex]\( h \)[/tex] of the triangle, we use the right triangle formed by drawing a perpendicular from the top vertex to the base. This splits the base into two equal parts of length [tex]\( \frac{b}{2} \)[/tex]:
[tex]\[ \frac{b}{2} = \frac{20.57142857142857}{2} \][/tex]
[tex]\[ \frac{b}{2} \approx 10.285714285714285 \][/tex]
Using the Pythagorean theorem in one of these right triangles:
[tex]\[ a^2 = \left(\frac{b}{2}\right)^2 + h^2 \][/tex]
Solving for [tex]\( h \)[/tex]:
[tex]\[ h^2 = a^2 - \left(\frac{b}{2}\right)^2 \][/tex]
[tex]\[ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} \][/tex]
[tex]\[ h = \sqrt{13.714285714285714^2 - 10.285714285714285^2} \][/tex]
[tex]\[ h \approx 9.071147352221454 \][/tex]
### Step 8: Find the area of the triangle
The area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
[tex]\[ A = \frac{1}{2} \times b \times h \][/tex]
[tex]\[ A = \frac{1}{2} \times 20.57142857142857 \times 9.071147352221454 \][/tex]
[tex]\[ A \approx 93.30322990856351 \][/tex]
### Summary of Results
- Length of each of the equal sides [tex]\( a \approx 13.714285714285714 \)[/tex] cm
- Length of the base [tex]\( b \approx 20.57142857142857 \)[/tex] cm
- Height of the triangle [tex]\( h \approx 9.071147352221454 \)[/tex] cm
- Area of the triangle [tex]\( A \approx 93.30322990856351 \)[/tex] cm²
