We will solve each right triangle separately. We are given more values from △ JKM than △ LKM. Therefore, we will start with △ JKM.
△ JKM
We will find the missing measures one at a time. In this case, this means that we want to find m ∠ JKM, KM, and JK.
Angle Measure
Let's analyze △ JKM.
To find m∠ JKM, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ JKM and m ∠ MJK add to 90.
m ∠ JKM + m ∠ MJK = 90
Since m∠ MJK= 41, we can substitute this value in our equation and find the measure of ∠ JKM.
m ∠ JKM + 41 = 90 ⇔ m ∠ JKM = 49
Side Lengths
Let x be the length of KM. We can find its value by using the tangent ratio.
The tangent of ∠ MJK is the ratio of the length of the leg opposite ∠ MJK to the length of its adjacent leg.
Tangent=Opposite/Adjacent ⇒ tan 41^(∘) =x/9
We can use the obtained equation to find x. We have to use a calculator to find the value of tan 41^(∘).
Therefore, the measure of KM is 7.8 ft, rounded to nearest tenth. Finally, we can find the measure of JK. To do it, we can use the Pythagorean Theorem.
a^2+ b^2 = c^2
For our triangle, the lengths of the legs are 7.8 and 9, and the hypotenuse is JK. We can substitute these values in the above equation, and solve for the missing side length.
We have found the measures of all the angles and the lengths of all the sides in △ JKM.
△ LKM
We will find the missing measures one at a time. In this case, this means that we will find m ∠ KLM, m ∠ MKL, and ML.
Angle Measures
Let's analyze △ LKM. We can find m ∠ KLM using the sine ratio.
The sine of ∠ KLM is the ratio of the length of the leg opposite ∠ KLM to the length of the hypotenuse.
Sine=Opposite/Hypotenuse ⇒ sin ∠ KLM =7.8/21
By definition, the inverse sine of 7.8 21 is the measure of ∠ KLM. To find it, we have to use a calculator.
To find m∠ MKL, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ MKL and m ∠ KLM add to 90.
m ∠ MKL + m ∠ KLM = 90
Now, we can substitute the measure of ∠ KLM in our equation and find the measure of ∠ MKL.
m ∠ MKL + 21.8 = 90 ⇔ m ∠ MKL ≈ 68.2
Side Length
Finally, we can find the length of ML. To do it, we can use the Pythagorean Theorem.
a^2+b^2=c^2
In this case, the lengths of one legs and the hypotenuse are 7.8 and 21, respectively. We want to find the length of the other leg, which is ML. Let's substitute these values in the above equation.
