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Vesnalui [34]
3 years ago
14

In isosceles right triangle ABC, point is on hypotenuse \overline{BC} such that \overline{AD} is an altitude of \triangle ABC an

d DC = 5. What is the area of triangle ABC?
Mathematics
1 answer:
Dmitriy789 [7]3 years ago
4 0

Answer:

Area of triangle is 25.

Step-by-step explanation:

We have been given an isosceles right triangle

Isosceles triangle is the triangle having two sides equal.

Figure is shown in attachment

By Pythagoras theorem

BC^2=AC^2+AB^2

AD is altitude which divides the triangle into two parts

DC=5 implies BC =10 since D equally divides BC

Let AC=a implies AB=a being Isosceles

On substituting the values in the Pythagoras theorem:

10^2=a^2+a^2

100=2a^2

\Rightarrow a^2=50

\Rightarrow a=\pm5\sqrt{2}

WE can find area of right triangle by considering height AB and AD

Area of triangle ABC is:

\frac{1}{2}\cdot BC\cdot AD     (1)

\Rightarrow \frac{1}{2}\cdot 10\cdot AD

And other method of area of triangle is:

\frac{1}{2}\cdot AB\cdot BC       (2)

Equating (1) and (2) we get:

\frac{1}{2}\cdot 10\cdot AD=\frac{1}{2}\cdot a\cdot a

\Rightarrow AD=\frac{a^2}{10}

\Rightarrow AD=\frac{50}{10}=5

Using area of triangle is: \frac{1}{2}\cdot BC\cdot AD

Now, the area of triangle ABC=\frac{1}{2}\cdot 5\cdot 10

\Rightarrow 25



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