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QuestionTrova il perimetro e l'area di un triangolo rettangolo con un cateto di 36 cm36 \mathrm{~cm} e l'altro cateto lungo 3/43/4 di esso.

Studdy Solution

STEP 1

Assumptions1. The triangle is a right triangle. One of the legs (cateto in Italian) measures36 cm3. The other leg is3/4 of the first leg4. We need to calculate the perimeter and the area of the triangle

STEP 2

First, we need to find the length of the other leg. We can do this by multiplying the length of the first leg by/4.
Other leg=First leg×4\text{Other leg} = \text{First leg} \times \frac{}{4}

STEP 3

Now, plug in the given value for the first leg to calculate the length of the other leg.
Other leg=36 cm×3\text{Other leg} =36\, \text{cm} \times \frac{3}{}

STEP 4

Calculate the length of the other leg.
Other leg=36 cm×34=27 cm\text{Other leg} =36\, \text{cm} \times \frac{3}{4} =27\, \text{cm}

STEP 5

Now that we have the lengths of the two legs, we can find the length of the hypotenuse using the Pythagorean theorem.
Hypotenuse=First leg2+Other leg2\text{Hypotenuse} = \sqrt{\text{First leg}^2 + \text{Other leg}^2}

STEP 6

Plug in the values for the first leg and the other leg to calculate the length of the hypotenuse.
Hypotenuse=362 cm2+272 cm2\text{Hypotenuse} = \sqrt{36^2\, \text{cm}^2 +27^2\, \text{cm}^2}

STEP 7

Calculate the length of the hypotenuse.
Hypotenuse=362 cm2+272 cm2=45 cm\text{Hypotenuse} = \sqrt{36^2\, \text{cm}^2 +27^2\, \text{cm}^2} =45\, \text{cm}

STEP 8

Now that we have the lengths of all three sides, we can calculate the perimeter of the triangle by adding up the lengths of the sides.
Perimeter=First leg+Other leg+Hypotenuse\text{Perimeter} = \text{First leg} + \text{Other leg} + \text{Hypotenuse}

STEP 9

Plug in the values for the first leg, the other leg, and the hypotenuse to calculate the perimeter.
Perimeter=36 cm+27 cm+45 cm\text{Perimeter} =36\, \text{cm} +27\, \text{cm} +45\, \text{cm}

STEP 10

Calculate the perimeter of the triangle.
Perimeter=36 cm+27 cm+45 cm=108 cm\text{Perimeter} =36\, \text{cm} +27\, \text{cm} +45\, \text{cm} =108\, \text{cm}

STEP 11

Now that we have the lengths of the two legs, we can calculate the area of the triangle using the formula for the area of a right triangle.
Area=×First leg×Other leg\text{Area} = \frac{}{} \times \text{First leg} \times \text{Other leg}

STEP 12

Plug in the values for the first leg and the other leg to calculate the area.
Area=2×36 cm×27 cm\text{Area} = \frac{}{2} \times36\, \text{cm} \times27\, \text{cm}

STEP 13

Calculate the area of the triangle.
Area=2×36 cm×27 cm=486 cm2\text{Area} = \frac{}{2} \times36\, \text{cm} \times27\, \text{cm} =486\, \text{cm}^2The perimeter of the triangle is108 cm and the area is486 cm².

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