Question

In ∆MNP, m∠N = 90º, NH – altitude, m∠P = 21º, PM = 4 cm. Find MH.

Answer 1

Answer:

0.51 cm

Step-by-step explanation:

In right triangle MNP, MP = 4 cm, m∠N = 90°, m∠P = 21°

By the sine definition,

$\sin \angle P=\dfrac{\text{Opposite leg}}{\text{Hypotenuse}}=\dfrac{MN}{MP}\\ \\MN=MP\sin \angle P\\ \\MN=4\sin 21^{\circ}\approx 1.43\ cm$

Now, consider right triangle HMN (it is right because NH is an altitude). By the cosine definition,

$\cos \angle M=\dfrac{\text{Adjacent leg}}{\text{Hypotenuse}}=\dfrac{MH}{MN}\\ \\MH=MN\cos \angle M$

In the right triangle, two acute angles are always complementary, so

$m\angle M=90^{\circ}-m\angle P=90^{\circ}-21^{\circ}=69^{\circ}$

Thus,

$MH=1.43\cos 69^{\circ}\approx 0.51\ cm$