Question

The sides of a triangle are 9cm, 12cm and 15cm.Find the length of the perpendicular drawn to the

side which is 15 cm.
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Answer 1

If 15^2 = 9^2 +12^2, the triangle satisfies the Pythagorean Theorem for a right triangle.

Is (15^2 = 81 + 144)?

Is (225 = 225)? Yes

Therefore the given triangle is a right triangle. Let the vertex of the right angle be C, the leg AC is 12, the hypotenuse AB is 15 and the leg BC is 9. The line perpendicular to the hypotenuse at a point D that connects to the vertex C is the "height" of the triangle ABC. It can be shown that the three angles of the triangle ABC are equal to those of triangle CBD. Thus triangle ABC is similar to CBD. The problem is to find the lenght of the line CD.

Using proportions of corresponding sides we have

CD/9 = 12/15

CD = 9*12/15

CD = 36/5

Answer: the line is 7.2 cm.

The hypotenuse side is 15 cm.

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Answer 2

Step-by-step explanation:

sides are 9 cm,  12 cm and 15 cm.

semi perimeter = (a + b + c ) / 2

                         = ( 9 + 12 + 15 ) / 2

                         = 36 / 2

                         = 18 cm

Area of the triangle = $\sqrt{s(s-a) (s-b) (s-c)}$                   [Heron's formula]

                                = $\sqrt{ 18 (18-9) (18-12) (18-15)}$

                                = √ ( 18 x 9 x 6 x 3 )

                               = √ ( 2 x 9 x 9 x 2 x 3 x 3)

                               = 2 x 9 x 3

                              = 54 cm²

Now,

Area of the triangle = 1 /2 x b x h               [ Pythagorean theorem]

                                 = 1 /2 x 15 x h = 54

                                 =>  h = ( 54 x 2 ) / ( 15 x 1 )

                                 =>  h =  108 / 15

                                 =>  h = 7.2 cm       ( Ans)