Question

In a right triangle ABC, CD is an altitude, such that AD=BC. Find AC, if AB=3 cm, and CD= 2 cm.

Answer 1

The length of AC will be 2.5495.... cm.

Step-by-step explanation

In the diagram below, ABC is a right angle triangle with altitude as CD.

So, triangle ADC will be also a right angle triangle, in which CD = 2 cm.

In triangle ADC, using Pythagorean theorem we will get.....

`AD^2 +CD^2= AC^2 \\ \\ AD^2+(2)^2= AC^2\\ \\ AD^2= AC^2 -4 \\ \\ AD= √(AC^2 -4) ..............................(1)`

Now in triangle ABC, using Pythagorean theorem.....

`AC^2 + BC^2= AB^2\\ \\ AC^2+ BC^2= (3)^2\\ \\ BC^2= 9-AC^2\\ \\ BC= √(9-AC^2)......................................(2)`

As it is given that AD = BC , so from equation (1) and (2) we will get.....

`√(AC^2 -4)=√(9-AC^2)\\ \\ AC^2-4= 9-AC^2\\ \\ 2AC^2= 9+4=13\\ \\ AC^2= (13)/(2)=6.5\\ \\ AC= √(6.5)=2.5495....`

So, the length of AC will be 2.5495.... cm.