In triangle ABC, AB = 8.9 m, Angle BAC = 90° and Angle ABC = 56°. H lies on BC such that AH
is perpendicular to BC. Find the length of
(i) AH
(ii) HC.

Please help me!! Thank you

(i) The length of AH is 7.3784 m

(ii) The length of HC is 10.9389 m

Step-by-step explanation:

The sum of the angles ABC, BAC and ACB is 180º, hence, ACB = 180-90-56 = 34º. Note also that AH breaks the rectangle triangle ABC into 2 rectangle triangles AHB and AHC.

i) In AHB, AB is the hypotenuse, and the angle ABH is equal to ABC, hence it is 56º. AH is the opposite side of that angle and we know that

Sin(ABH) = opposite/hypotenuse = AH/8.9

Thus,

AH = 8.9*Sin(ABH) = 8.9*Sin(56º) = 7.3784 m

ii) In AHC, AH is also the opposite of the angle ACH (which meassures 34º), and HC is the adjacent of that angle (the hypotenuse is AC). As a result

Tan(ACH) = Tan(34º) = 0.6745 = opposite/adjacent = AH/HC = 7.3784/HC

Hence,

HC = 7.3785/0.6745 = 10.9389 m

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