Question

3 practice 3
triangle abc has a right angle at c.

select all measurements which would mean it has a hypotenuse with a length of 10 units.
a angle a measures 20°, and segment bc is 2 units long.
b segment ac is 7 units long, and segment bc is 3 units long.
c angle b measures 50°, and segement bc is 4 units long.
d angle a measures 30°, and segment bc is 5 units long.
e segment ac is 8 units long, and segment bc is 6 units long.

Explanation:

Step1: Recall Pythagorean theorem and trigonometric ratios

For a right triangle with right angle at \( C \), hypotenuse \( AB \), legs \( AC \) and \( BC \):

• Pythagorean theorem: \( AB^{2}=AC^{2} + BC^{2} \)
• Trigonometric ratios: \( \sin A=\frac{BC}{AB} \), \( \sin B=\frac{AC}{AB} \), \( \cos A=\frac{AC}{AB} \), \( \cos B=\frac{BC}{AB} \)

Step2: Analyze Option A

Given \( \angle A = 20^{\circ} \), \( BC = 2 \) units.
Using \( \sin A=\frac{BC}{AB} \), so \( AB=\frac{BC}{\sin A}=\frac{2}{\sin20^{\circ}}\approx\frac{2}{0.3420}\approx5.85
eq 10 \). So A is incorrect.

Step3: Analyze Option B

Given \( AC = 7 \) units, \( BC = 3 \) units.
Using Pythagorean theorem: \( AB^{2}=7^{2}+3^{2}=49 + 9=58 \), \( AB=\sqrt{58}\approx7.62
eq10 \). So B is incorrect.

Step4: Analyze Option C

There is a typo (segement instead of segment), but given \( \angle B = 50^{\circ} \), \( BC = 4 \) units.
Using \( \cos B=\frac{BC}{AB} \), so \( AB=\frac{BC}{\cos B}=\frac{4}{\cos50^{\circ}}\approx\frac{4}{0.6428}\approx6.22
eq10 \). So C is incorrect.

Step5: Analyze Option D

Given \( \angle A = 30^{\circ} \), \( BC = 5 \) units.
Using \( \sin A=\frac{BC}{AB} \), \( \sin30^{\circ}=\frac{1}{2} \), so \( AB=\frac{BC}{\sin A}=\frac{5}{\frac{1}{2}} = 10 \). So D is correct.

Step6: Analyze Option E

Given \( AC = 8 \) units, \( BC = 6 \) units.
Using Pythagorean theorem: \( AB^{2}=8^{2}+6^{2}=64 + 36 = 100 \), so \( AB = 10 \). So E is correct.

Answer:

D. Angle \( A \) measures \( 30^{\circ} \), and segment \( BC \) is 5 units long.
E. Segment \( AC \) is 8 units long, and segment \( BC \) is 6 units long.