Question

solve the given right triangle for its missing angle and side measures.
note: figure not drawn to scale
a. ( mangle d = 55^circ ), ( de approx 4.40 ) units, ( df approx 13.85 ) units
b. ( mangle d = 35^circ ), ( de approx 8.40 ) units, ( df approx 13.85 ) units
c. ( mangle d = 35^circ ), ( de approx 8.40 ) units, ( df approx 14.65 ) units
d. ( mangle d = 55^circ ), ( de approx 8.40 ) units, ( df approx 14.85 ) units

Explanation:

Step1: Find m∠D

In right triangle DEF, ∠E=90°, ∠F=35°. Sum of angles: 180°-90°-35°=55°. So m∠D=55°.

Step2: Calculate DE

tan(∠F)=DE/FE → DE=FE×tan(35°)=12×tan(35°)≈12×0.7002≈8.40 units.

Step3: Calculate DF

cos(∠F)=FE/DF → DF=FE/cos(35°)=12/cos(35°)≈12/0.8192≈14.65 units? Wait, correction: Wait, cos(35°)≈0.8192, 12/0.8192≈14.65? No, wait 12/cos(35)≈12/0.81915≈14.65, but option D says DF≈14.05? Wait no, wait: sin(35°)=DE/DF → DF=DE/sin(35°)=8.40/0.5736≈14.64≈14.65? Wait option C says DF≈14.65, but m∠D=35° which is wrong. Wait no, m∠D=55°, so cos(∠D)=DE/DF → DF=DE/cos(55°)=8.40/0.5736≈14.64≈14.65? Wait option D: m∠D=55°, DE≈8.40, DF≈14.05? Wait maybe calculation: FE=12, ∠F=35°, so DF=12/cos(35°)≈12/0.819≈14.65, DE=12×tan(35°)≈8.40, m∠D=55°. So option D: DF≈14.05? Wait no, 12/cos(35)≈14.65, but option D says 14.05? Wait maybe typo, but option D has correct angle m∠D=55°, DE≈8.40, which matches. The closest is D (maybe DF is 14.65 but written as 14.05 by mistake, but D is the only one with m∠D=55° and DE≈8.40). So answer is D.

Answer:

D. m∠D = 55°, DE ≈ 8.40 units, DF ≈ 14.05 units