Question

In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths. AB=9, and AC=12, find HC.

Answer 1

Answer:

Step-by-step explanation:

It is given that in △ABC, ∠ABC=90°, BH is an altitude and AB=9 and AC=12, thus using the Pythagoras theorem in △ABC, we get

$(AC)^{2}=(AB)^{2}+(BC)^{2}$

$(12)^{2}=(9)^2+(BC)^2$

$144=81+(BC)^2$

$(BC)^2=63$

$BC=\sqrt{63}$

Now, From ΔABC and ΔHBC, we have

∠ABC=∠BHC(each90)

∠ACB=∠HCB (Common)

By AA similarity, ΔABC is similar to ΔHBC.

Thus, using the similarity condition, we get

$\frac{HC}{BC}=\frac{BC}{AC}$

$\frac{HC}{\sqrt{63}}=\frac{\sqrt{63}}{12}$

$HC=\frac{63}{12}=\frac{21}{4}$