Let D be the Intersection of Height of the Triangle and Base of the Triangle BC
From the Figure, We can notice that Triangle ADB is a Right angled Triangle.
We know that, In a Right angled Triangle :
(Hypotenuse)² = (First Leg)² + (Second leg)²
In Triangle, ADB : AB is the Hypotenuse, AD is the First leg and BD is the Second leg
Given : AB = 15 and BD = 9
Substituting the values, We get :
(15)² = (AD)² + (9)²
225 = (AD)² + 81
(AD)² = 225 - 81
(AD)² = 144
(AD)² = (12)²
AD = 12
We know that, In a Right angled Triangle :
![\bigstar\;\;\boxed{\mathsf{Tan\theta = \dfrac{Opposite\;Side}{Adjacent\;Side}}}](https://tex.z-dn.net/?f=%5Cbigstar%5C%3B%5C%3B%5Cboxed%7B%5Cmathsf%7BTan%5Ctheta%20%3D%20%5Cdfrac%7BOpposite%5C%3BSide%7D%7BAdjacent%5C%3BSide%7D%7D%7D)
Now, Consider Triangle ADC : With respect to 45° Angle, AD is the Opposite Side and DC is the Adjacent Side
![:\implies \mathsf{In\;Triangle\;ADC,\;Tan45^{\circ} = \dfrac{AD}{DC}}](https://tex.z-dn.net/?f=%3A%5Cimplies%20%5Cmathsf%7BIn%5C%3BTriangle%5C%3BADC%2C%5C%3BTan45%5E%7B%5Ccirc%7D%20%3D%20%5Cdfrac%7BAD%7D%7BDC%7D%7D)
![\mathsf{:\implies 1 = \dfrac{12}{DC}}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3A%5Cimplies%201%20%3D%20%5Cdfrac%7B12%7D%7BDC%7D%7D)
DC = 12
Total Length of the Base (BC) = BD + DC
Total Length of the Base (BC) = 9 + 12
Total Length of the Base (BC) = 21
We know that, Area of the Triangle is given by :
![\bigstar\;\;\boxed{\mathsf{Area = \dfrac{1}{2} \times Base \times Height}}](https://tex.z-dn.net/?f=%5Cbigstar%5C%3B%5C%3B%5Cboxed%7B%5Cmathsf%7BArea%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20Base%20%5Ctimes%20Height%7D%7D)
In Triangle, ABC : AD is the Height and BC is the Base
![:\implies \mathsf{Area\;of\;the\;Triangle\;ABC = \dfrac{1}{2} \times BC \times AD}](https://tex.z-dn.net/?f=%3A%5Cimplies%20%5Cmathsf%7BArea%5C%3Bof%5C%3Bthe%5C%3BTriangle%5C%3BABC%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20BC%20%5Ctimes%20AD%7D)
![:\implies \mathsf{Area\;of\;the\;Triangle\;ABC = \dfrac{1}{2} \times 21 \times 12}](https://tex.z-dn.net/?f=%3A%5Cimplies%20%5Cmathsf%7BArea%5C%3Bof%5C%3Bthe%5C%3BTriangle%5C%3BABC%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%2021%20%5Ctimes%2012%7D)
![:\implies \mathsf{Area\;of\;the\;Triangle\;ABC = (21 \times 6)}](https://tex.z-dn.net/?f=%3A%5Cimplies%20%5Cmathsf%7BArea%5C%3Bof%5C%3Bthe%5C%3BTriangle%5C%3BABC%20%3D%20%2821%20%5Ctimes%206%29%7D)
![:\implies \mathsf{Area\;of\;the\;Triangle\;ABC = 126}](https://tex.z-dn.net/?f=%3A%5Cimplies%20%5Cmathsf%7BArea%5C%3Bof%5C%3Bthe%5C%3BTriangle%5C%3BABC%20%3D%20126%7D)