Question

déterminer la mesure des côtés manquants 1. déterminez la mesure du côté, x, dans chacun des triangles rectangles suivants :(collection cec ) a) hypotenuse 24 cm 33° x b) a 12 cm x hypotenuse 50° c) x 29° 15 cm d) 6 cm 18° x e) hypotenuse 61° 55 cm x f) x 51° 18 cm

Explanation:

Part (a)

Step 1: Identify trigonometric ratio

We have a right triangle with hypotenuse \( 24 \, \text{cm} \), angle \( 33^\circ \), and opposite side \( x \). So we use \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \).
\( \sin(33^\circ)=\frac{x}{24} \)

Step 2: Solve for \( x \)

Multiply both sides by \( 24 \): \( x = 24\times\sin(33^\circ) \)
Calculate \( \sin(33^\circ)\approx0.5446 \), so \( x\approx24\times0.5446\approx13.07 \, \text{cm} \)

Step 1: Identify trigonometric ratio

We have a right triangle with adjacent side \( 12 \, \text{cm} \), angle \( 50^\circ \), and hypotenuse \( x \). So we use \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \).
\( \cos(50^\circ)=\frac{12}{x} \)

Step 2: Solve for \( x \)

Rearrange: \( x=\frac{12}{\cos(50^\circ)} \)
Calculate \( \cos(50^\circ)\approx0.6428 \), so \( x\approx\frac{12}{0.6428}\approx18.67 \, \text{cm} \)

Step 1: Identify trigonometric ratio

We have a right triangle with hypotenuse \( 15 \, \text{cm} \), angle \( 29^\circ \), and adjacent side \( x \). So we use \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \).
\( \cos(29^\circ)=\frac{x}{15} \)

Step 2: Solve for \( x \)

Multiply both sides by \( 15 \): \( x = 15\times\cos(29^\circ) \)
Calculate \( \cos(29^\circ)\approx0.8746 \), so \( x\approx15\times0.8746\approx13.12 \, \text{cm} \)

Answer:

\( x\approx13.07 \, \text{cm} \)

Part (b)