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leva [86]
2 years ago
11

Let ABC be a triangle such that AB=13 BC=14 and CA=15. D is a point on BC such that AD Bisects

Mathematics
1 answer:
Yuri [45]2 years ago
5 0

Answer:

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 x ; hence, DC will be 14-x

and AD be y

To, find the area of triangle ADC

Area of triangle ADC  = \frac{1}{2} (DC)(AD)

= \frac{1}{2}(14-x)(y)

We will have to determine x and y

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

Area =\sqrt{s(s-a)(s-b)(s-c)}

Where s = \frac{a+b+c}{2}

For the given triangle ABC

Let a = AB, b = BC, and c = CA

Hence, a = 13, b= 14, and c = 15

∴ s = \frac{13+14+15}{2} \\s= \frac{42}{2}\\s = 21

Then,

Area of triangle ABC = \sqrt{(21)(21-13)(21-14)(21-15)}

Area of triangle ABC = \sqrt{(21)(8)(7)(6)} = \sqrt{7056}

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)

Area of triangle ADB = \frac{1}{2}(x)(y)

Hence,

Area of triangle ABC =  \frac{1}{2}(x)(y) + \frac{1}{2}(14-x)(y)

84 =   \frac{1}{2}(x)(y) + \frac{1}{2}(14-x)(y)

∴ 84 = \frac{1}{2}(xy) + 7y - \frac{1}{2}(xy)

84 = 7y\\y = \frac{84}{7}

∴ y = 12

Hence, y = 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}

/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)

Area of triangle ADC  = \frac{1}{2} (9)(12)

Area of triangle ADC  = 9 × 6

Area of triangle ADC  = 54 square unit

Hence, Area of triangle ADC  is 54 square unit.

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