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tyliahhayes
02.09.2020 •
Mathematics
Let ABC be a triangle such that AB=13 BC=14 and CA=15. D is a point on BC such that AD Bisects
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Ответ:
Area of triangle ADC is 54 square unit
Step-by-step explanation:
Here is the complete question:
Let ABC be a triangle such that AB=13, BC=14, and CA=15. D is a point on BC such that AD bisects angle A. Find the area of triangle ADC .
Step-by-step explanation:
Please see the attachment below for an illustrative diagram
Considering the diagram,
BC = BD + DC = 14
Let BD be
; hence, DC will be ![14-x](/tpl/images/0737/4814/cec79.png)
and AD be![y](/tpl/images/0737/4814/9512c.png)
To, find the area of triangle ADC
Area of triangle ADC =![\frac{1}{2} (DC)(AD)](/tpl/images/0737/4814/0cdbf.png)
=![\frac{1}{2}(14-x)(y)](/tpl/images/0737/4814/d9d59.png)
We will have to determine
and ![y](/tpl/images/0737/4814/9512c.png)
First we will find the area of triangle ABC
The area of triangle ABC can be determined using the Heron's formula.
Given a triangle with a,b, and c
Where![s = \frac{a+b+c}{2}](/tpl/images/0737/4814/a7baa.png)
For the given triangle ABC
Let
= AB,
= BC, and
= CA
Hence,
and ![c = 15](/tpl/images/0737/4814/7f090.png)
∴![s = \frac{13+14+15}{2} \\s= \frac{42}{2}\\s = 21](/tpl/images/0737/4814/6ba87.png)
Then,
Area of triangle ABC =![\sqrt{(21)(21-13)(21-14)(21-15)}](/tpl/images/0737/4814/055af.png)
Area of triangle ABC =
= ![\sqrt{7056}](/tpl/images/0737/4814/4c903.png)
Area of triangle ABC = 84 square unit
Now, considering the diagram
Area of triangle ABC = Area of triangle ADB + Area of triangle ADC
Area of triangle ADB =![\frac{1}{2} (BD)(AD)](/tpl/images/0737/4814/9e8f8.png)
Area of triangle ADB =![\frac{1}{2}(x)(y)](/tpl/images/0737/4814/70bb9.png)
Hence,
Area of triangle ABC =
+ ![\frac{1}{2}(14-x)(y)](/tpl/images/0737/4814/d9d59.png)
84 =
+ ![\frac{1}{2}(14-x)(y)](/tpl/images/0737/4814/d9d59.png)
∴![84 = \frac{1}{2}(xy) + 7y - \frac{1}{2}(xy)](/tpl/images/0737/4814/2c145.png)
∴![y = 12](/tpl/images/0737/4814/44715.png)
Hence,
AD = 12
Now, we can find BD
Considering triangle ADB,
From Pythagorean theorem,
/AB/² = /AD/² + /BD/²
∴13² = 12² + /BD/²
/BD/² = 169 - 144
/BD/ =![\sqrt{25}](/tpl/images/0737/4814/3e941.png)
/BD/ = 5
But, BD + DC = 14
Then, DC = 14 - BD = 14 - 5
BD = 9
Now, we can find the area of triangle ADC
Area of triangle ADC =![\frac{1}{2} (DC)(AD)](/tpl/images/0737/4814/0cdbf.png)
Area of triangle ADC =![\frac{1}{2} (9)(12)](/tpl/images/0737/4814/ee308.png)
Area of triangle ADC = 9 × 6
Area of triangle ADC = 54 square unit
Hence, Area of triangle ADC is 54 square unit.
Ответ:
11) obtuse
13)right
15) acute