Question

construction
raisonner • représenter d1.3
avec ces seules mesures, naël prétend pouvoir donner une approximation de la hauteur totale de sa maison.
que peut - on en penser ?
abcd est un rectangle.

Explanation:

Step1: Analyze Triangle EAD

In triangle \( EAD \), \( EA = ED \) (marked as equal), so it's isosceles. The vertex angle \( \angle E = 132^\circ \). The base angles \( \angle EAD \) and \( \angle EDA \) are equal. Using the angle - sum property of a triangle (\( \text{Sum of angles in a triangle}=180^\circ \)), we calculate each base angle:
\( \angle EAD=\angle EDA=\frac{180 - 132}{2}=\frac{48}{2} = 24^\circ \)

Step2: Analyze Rectangle ABCD

Since \( ABCD \) is a rectangle, \( AD = BC = 9 \) (opposite sides of a rectangle are equal), and \( \angle ABC = 90^\circ \). Also, \( \angle ADB=\angle DBC = 24^\circ \) (alternate - interior angles for \( AD\parallel BC \) and transversal \( BD \))

Step3: Analyze Triangle ABD (or CBD)

In right - triangle \( ABD \) (since \( \angle ABC = 90^\circ \)), we can find \( AB \) (height of the rectangle) using trigonometry. \( \tan(24^\circ)=\frac{AB}{AD} \), so \( AB = AD\times\tan(24^\circ)=9\times\tan(24^\circ) \)

Step4: Analyze Triangle EAD for height from E to AD

Let \( h_1 \) be the height from \( E \) to \( AD \) (in triangle \( EAD \)). In the isosceles triangle \( EAD \), if we drop a perpendicular from \( E \) to \( AD \), it bisects \( AD \) and the vertex angle. Let the foot of the perpendicular be \( O \). Then \( \angle EAO = 24^\circ \), and \( AO=\frac{AD}{2}=\frac{9}{2} = 4.5 \). Using trigonometry, \( \tan(24^\circ)=\frac{h_1}{AO} \), so \( h_1=AO\times\tan(24^\circ)=4.5\times\tan(24^\circ) \)

Step5: Total Height of the House

The total height of the house \( H=AB + h_1 \). Since \( AB = 9\times\tan(24^\circ) \) and \( h_1 = 4.5\times\tan(24^\circ) \), we can calculate \( H=(9 + 4.5)\times\tan(24^\circ)=13.5\times\tan(24^\circ) \)
We know that \( \tan(24^\circ)\approx0.4452 \), so \( H\approx13.5\times0.4452\approx6.01 \) (approximate value). Since we can use the given angle (\( 24^\circ \)) and the length of \( BC = 9 \) (which is equal to \( AD \)) to find the heights of the rectangular part and the triangular part of the roof, Naël can indeed give an approximation of the total height of the house.

Answer:

Naël can give an approximation of the total height of the house. By using the properties of isosceles triangles, rectangles, and trigonometric ratios (tangent of \( 24^\circ \)) with the given length (\( BC = 9 \)) and angles (\( 24^\circ \) and \( 132^\circ \)), we can calculate the height of the rectangular part (\( AB \)) and the height of the triangular roof part (from \( E \) to \( AD \)) and sum them to get the total height. For example, if we calculate \( \tan(24^\circ)\approx0.4452 \), the total height \( H\approx13.5\times0.4452\approx6.01 \) (the value depends on the precision of \( \tan(24^\circ) \) used).