Question

What is the value of x in the figure below? In this diagram, AABD - ACAD.АXB5 D9CA. 126O B. 45C. 45O D.914E. 170F. 14

Answer 1

We are given a right angled triangle that also houses two other right angles and are asked to find the length of the line opposite to acute angles in the two inner right angled triangles.

Our approach is to create a relationship for one of the lengths and in two instances and equate the length to allow us get our variable.

We will employ the Pythagoras Theorem that states that the square of the length of the hypothenuse is a sum of the length of the square of the opposite and the square of the adjacent.

`\begin{gathered} \text{Length AC = }\sqrt[]{9^2+x^2} \\ \text{Length AB = }\sqrt[]{5^2+x^2} \\ \text{Length AC = }\sqrt{\text{14}^2-(\sqrt[]{5^2+x^2})^2} \end{gathered}`

We can equate the 2 lengths of the square of AC.

`\begin{gathered} AC^2=\text{ (}\sqrt[]{9^2+x^2})^2=(\text{ }\sqrt[]{\text{14}^2-(\sqrt[]{5^2+x^2})^2})^2 \\ AC^2=9^2+x^2=14^2-5^2-x^2 \\ \text{Add x}^2\text{ to both sides to get:} \\ 2x^2+9^2=14^2-5^2 \\ Subtract9^2\text{ from both sides to get:} \\ 2x^2=14^2-5^2-9^2 \\ 2x^2=90 \\ \text{Divide both sides by 2 to get:} \\ x^2=45 \\ \text{Getting the square root of both sides gives:} \\ √(x^2)=√(45) \\ x=\sqrt[]{45} \end{gathered}`

OPTION C