The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy the relationship $T=R^2$. If the sides of the rectangle are in the ratio 1: 3, then the length, in cm, of the longer side of the rectangle, is

The correct answer is 27 cm.

Let us carefully define our variables and work through this problem step by step.

Step 1: Define Variables

Let the side of the equilateral triangle be a cm.
Since the sides of the rectangle are in the ratio 1 : 3, let the shorter side be x cm and the longer side be 3x cm.

Step 2: Apply the Perimeter Condition

Perimeter of equilateral triangle $=3a$

Perimeter of rectangle $=2(x+3x)=8x$

Their sum is 90 cm:

$3a+8x=90⟹a=\frac{90-8x}{3}\cdots \left(1\right)$

Step 3: Write the Area Expressions

Area of the equilateral triangle:

$T=\frac{\sqrt{3}}{4}{a}^2$

Area of the rectangle:

$R=x\times 3x=3x^2$

Step 4: Apply the Area Relationship

The given relationship between the two areas is:

$R=T^2$

Substituting the expressions for T and R:

$3x^2={\left(\frac{\sqrt{3}}{4}{a}^2\right)}^2=\frac{3}{16}{a}^4\cdots \left(2\right)$

Step 5: Substitute and Solve

Substitute $a=\frac{90-8x}{3}$ from equation (1) into equation (2):

$3x^2=\frac{3}{16}\times {\left(\frac{90-8x}{3}\right)}^4$

Multiply both sides by $\frac{16}{3}$:

$16x^2=\frac{{(90-8x)}^4}{81}$

Now let us test the answer option $x=9$ (i.e., shorter side = 9 cm, longer side = 27 cm):

From equation (1):

$a=\frac{90-8\left(9\right)}{3}=\frac{90-72}{3}=\frac{18}{3}=6\text{cm}$

Step 6: Verify with the Area Relationship

Compute the triangle area T:

$T=\frac{\sqrt{3}}{4}\times 6^2=\frac{\sqrt{3}}{4}\times 36=9\sqrt{3}\text{sq cm}$

Compute $T^2$:

$T^2={\left(9\sqrt{3}\right)}^2=81\times 3=243$

Compute the rectangle area R:

$R=x\times 3x=9\times 27=243\text{sq cm}$

Check:

$R=T^2⟹243=243✓$

Step 7: Confirm the Perimeter

$3a+8x=3\left(6\right)+8\left(9\right)=18+72=90\text{cm}✓$

Conclusion:
The shorter side of the rectangle is 9 cm and the longer side is 3 × 9 = 27 cm. All conditions are perfectly satisfied, confirming that the length of the longer side of the rectangle is 27 cm.