Option d: Angle A is 30 degrees, BC is 5 units
Option e: AC is 8 units, BC is 6 units
Explanation:
The triangle ABC has a right angle at C.
The length of the hypotenuse is 10 units.
The image of the triangle with this measurement is attached below:
Option a: Angle A is 20 degrees, BC is 2 units
![\begin{aligned}\sin 20 &=\frac{2}{h y p} \\h y p &=\frac{2}{\sin 20} \\&=5.8476\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Csin%2020%20%26%3D%5Cfrac%7B2%7D%7Bh%20y%20p%7D%20%5C%5Ch%20y%20p%20%26%3D%5Cfrac%7B2%7D%7B%5Csin%2020%7D%20%5C%5C%26%3D5.8476%5Cend%7Baligned%7D)
Since, hypotenuse is 10 units, Option a is not correct answer.
Option b: AC is 7 units, BC is 3 units
![\begin{aligned}A B &=\sqrt{7^{2}+3^{2}} \\&=\sqrt{49+9} \\&=\sqrt{58}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7DA%20B%20%26%3D%5Csqrt%7B7%5E%7B2%7D%2B3%5E%7B2%7D%7D%20%5C%5C%26%3D%5Csqrt%7B49%2B9%7D%20%5C%5C%26%3D%5Csqrt%7B58%7D%5Cend%7Baligned%7D)
Since, hypotenuse is 10 units, Option b is not correct answer.
Option c: Angle B is 50 degrees, BC is 4 units
![\begin{aligned}\cos 50 &=\frac{4}{h y p} \\h y p &=\frac{4}{\cos 50} \\&=6.222\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Ccos%2050%20%26%3D%5Cfrac%7B4%7D%7Bh%20y%20p%7D%20%5C%5Ch%20y%20p%20%26%3D%5Cfrac%7B4%7D%7B%5Ccos%2050%7D%20%5C%5C%26%3D6.222%5Cend%7Baligned%7D)
Since, hypotenuse is 10 units, Option c is not correct answer.
Option d: Angle A is 30 degrees, BC is 5 units
![\begin{aligned}\sin 30 &=\frac{5}{h y p} \\h y p &=\frac{5}{\sin 30} \\&=10\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Csin%2030%20%26%3D%5Cfrac%7B5%7D%7Bh%20y%20p%7D%20%5C%5Ch%20y%20p%20%26%3D%5Cfrac%7B5%7D%7B%5Csin%2030%7D%20%5C%5C%26%3D10%5Cend%7Baligned%7D)
Since, hypotenuse is 10 units, Option d is the correct answer.
Option e: AC is 8 units, BC is 6 units
![\begin{aligned}A B &=\sqrt{8^{2}+6^{2}} \\&=\sqrt{64+36} \\&=\sqrt{100} \\&=10\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7DA%20B%20%26%3D%5Csqrt%7B8%5E%7B2%7D%2B6%5E%7B2%7D%7D%20%5C%5C%26%3D%5Csqrt%7B64%2B36%7D%20%5C%5C%26%3D%5Csqrt%7B100%7D%20%5C%5C%26%3D10%5Cend%7Baligned%7D)
Since, hypotenuse is 10 units, Option e is the correct answer.
Thus, Option d and e are the correct answers.