Answer:
BC = 35 m
AD = 16.8 m
Step-by-step explanation:
Since triangle ABC is a right triangle, we can use the Pythagorean Theorem to find the length of its hypotenuse (BC).
[tex]\boxed{\begin{array}{l}\underline{\textsf{Pythagorean Theorem}}\\\\a^2+b^2=c^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}[/tex]
In this case:
- a = AB = 28 m
- b = AC = 21 m
- c = BC
Substitute the values into the formula and solve for BC:
[tex]AB^2+AC^2=BC^2\\\\28^2+21^2=BC^2\\\\784+441=BC^2\\\\BC^2=1225\\\\BC=\sqrt{1225}\\\\BC=35[/tex]
Therefore, the length of BC is 35 meters.
[tex]\dotfill[/tex]
The Geometric Mean Theorem (Leg Rule) states that if an altitude is drawn from the right angle of a right triangle to its hypotenuse, it divides the hypotenuse into two segments. The ratio of the length of the hypotenuse to one of the legs is equal to the ratio of that leg to the segment of the hypotenuse adjacent to it.
[tex]\boxed{\sf \dfrac{Hypotenuse}{Leg\:1}=\dfrac{Leg\:1}{Segment\;1}}\quad \sf and \quad \boxed{\sf \dfrac{Hypotenuse}{Leg\:2}=\dfrac{Leg\:2}{Segment\;2}}[/tex]
In this case:
- Leg 1 = AC = 21 m
- Leg 2 = AB = 29 m
- Segment 1 = CD
- Segment 2 = BD
- Hypotenuse = BC = 35 m
Substitute the hypotenuse, leg 1 and segment 1 into the first equation and solve for CD:
[tex]\dfrac{35}{21}=\dfrac{21}{CD} \\\\\\CD=\dfrac{21\cdot 21}{35}\\\\\\ CD=12.6[/tex]
Now, use the Pythagorean Theorem again with right triangle ACD to find the length of AD:
[tex]AD^2 + CD^2 = AC^2\\\\AD^2 + 12.6^2= 21^2\\\\AD^2 + 158.76= 441\\\\AD^2=282.24\\\\AD=\sqrt{282.24}\\\\AD=16.8[/tex]
Therefore, the length of AD is 16.8 meters.
