Question

Taina, Luther, and Della are tapping a balloon to each other in the air, trying to keep it from touching the ground. The distance from Taina to Luther is 22 inches. The distance from Luther to Della is 40 inches. The distance from Della to Taina is 34 inches. Find the measures of the three angles in the triangle. Select one: a. m∠L=58.1m∠L=58.1, m∠T=88.5m∠T=88.5, m∠D=33.4m∠D=33.4 b. m∠L=33.4m∠L=33.4, m∠T=58.1m∠T=58.1, m∠D=88.5m∠D=88.5 c. m∠L=88.5m∠L=88.5, m∠T=33.4m∠T=33.4, m∠D=58.1m∠D=58.1 d. m∠L=60m∠L=60, m∠T=60m∠T=60, m∠D=60

Answer 1

a. m∠L=58.1, m∠T=88.5, m∠D=33.4

Explanation:

Suppose T, L and D represents Taina, Luther, and Della respectively,

According to the question,

TD = 34 inches,

TL = 22 inches,

DL = 40 inches,

By the law of cosine,

`DL^2 = TD^2 + TL^2 - 2* TD* TL cos T`

`2* TD* TL cos T= TD^2 + TL^2-DL^2`

`\implies cos T = (TD^2 + TL^2-DL^2)/(2* TD* TL)`

By substituting values,

`cos T = ( 34^2+22^2-40^2)/(2* 34* 22)`

`cos T = (1156+484-1600)/(1496)`

`cos T=(40)/(1496)`

`\implies m\angle T=cos^(-1)((40)/(1496))=88.4678446876\approx 88.5^(\circ)`

Similarly,

`cos L = (DL^2 + TL^2-TD^2)/(2* DL* TL)`

`cos L=(40^2+22^2-34^2)/(2* 40* 22)`

`cosL=(928)/(1760)`

`m\angle L=cos^(-1)((928)/(1760))=58.1786314749\approx 58.1^(\circ)`

`cos D = (DL^2 + TD^2-TL^2)/(2* DL* TD)`

`cos D=(40^2+34^2-22^2)/(2* 40* 34)`

`cosD=(2272)/(2720)`

`m\angle D=cos^(-1)((2272)/(2720))=33.3535238375\approx 33.4^(\circ)`

Hence, option 'a' is correct.

Answer 2

a. m∠L=58.1m∠L=58.1, m∠T=88.5m∠T=88.5, m∠D=33.4m∠D=33.4