Question

In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths. AH=4 and HC=1, find BH.

Answer 1

Answer:

$BH=2$

Step-by-step explanation:

Please find the attachment.

We have been given that triangle ABC is a right triangle, having a right angle at point B and BH is the altitude.

We can see from our attachment that the altitude BH is drawn to hypotenuse AC.

Altitude geometric mean theorem states that the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Each leg of the right triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Using the above theorem we can set proportions for our given side lengths as:

$\frac{\text{Left}}{\text{Altitude}}=\frac{\text{Altitude}}{\text{Right}}$      

Upon substituting our given values we will get,

$\frac{1}{BH}=\frac{BH}{4}$

Upon cross multiplying our equation we will get,

$BH*BH=1*4$

$(BH)^2=4$  

Taking square root of both sides of our equation we will get,

$BH=\sqrt{4}$

$BH=2$

Therefore, BH is equals to 2 units.