Question

un triangle a une aire de 14,7 dm² et une base de 8,4 dm. un trapèze a une aire de 20,72 dm² et une hauteur de 2,8 dm. lune de ses bases mesure 11 dm. coche chaque énoncé vrai parmi les suivants. a) la plus longue base du trapèze mesure 11 dm. b) la hauteur du trapèze est supérieure à la hauteur du triangle. c) la hauteur du triangle est inférieure à la mesure de la petite base du trapèze.

Explanation:

To solve this, we first find the height of the triangle and the other base of the trapezoid.

Step 1: Find the height of the triangle

The formula for the area of a triangle is \( A = \frac{1}{2} \times b \times h \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height. We know \( A = 14.7 \, \text{dm}^2 \) and \( b = 8.4 \, \text{dm} \). We solve for \( h \):

$$ 14.7 = \frac{1}{2} \times 8.4 \times h $$

First, simplify \( \frac{1}{2} \times 8.4 = 4.2 \):

$$ 14.7 = 4.2 \times h $$

Then, divide both sides by 4.2:

$$ h = \frac{14.7}{4.2} = 3.5 \, \text{dm} $$

Step 2: Find the other base of the trapezoid

The formula for the area of a trapezoid is \( A = \frac{(b_1 + b_2)}{2} \times h \), where \( A \) is the area, \( b_1 \) and \( b_2 \) are the bases, and \( h \) is the height. We know \( A = 20.72 \, \text{dm}^2 \), \( h = 2.8 \, \text{dm} \), and one base \( b_1 = 11 \, \text{dm} \). We solve for \( b_2 \):

$$ 20.72 = \frac{(11 + b_2)}{2} \times 2.8 $$

First, divide both sides by 2.8:

$$ \frac{20.72}{2.8} = \frac{(11 + b_2)}{2} $$

Calculate \( \frac{20.72}{2.8} = 7.4 \):

$$ 7.4 = \frac{(11 + b_2)}{2} $$

Multiply both sides by 2:

$$ 14.8 = 11 + b_2 $$

Subtract 11 from both sides:

$$ b_2 = 14.8 - 11 = 3.8 \, \text{dm} $$

Now we analyze each statement:

Statement a)

The bases of the trapezoid are 11 dm and 3.8 dm. The longer base is 11 dm, so this statement is true.

Statement b)

The height of the trapezoid is 2.8 dm, and the height of the triangle is 3.5 dm. Since \( 2.8 < 3.5 \), the height of the trapezoid is not superior to the height of the triangle. This statement is false.

Statement c)

The height of the triangle is 3.5 dm, and the smaller base of the trapezoid is 3.8 dm. Since \( 3.5 < 3.8 \), the height of the triangle is inferior to the smaller base of the trapezoid. This statement is true.

Answer:

a) La plus longue base du trapèze mesure 11 dm.
c) La hauteur du triangle est inférieure à la mesure de la petite base du trapèze.