Question

Triangle has sides 10cm,17cm,21cm.A square is inscribed in the triangle one side of the square lies on the longest side of the triangle the other two vertices of the square touch the two shorter sides of the triangle the length of the side of the square is

Answer 1

The length of the side of the square is 5.8 cm

Given that;

Sides of the triangle A= 1Ocm, B=17cm and C=21cm

To find;

The length of the side of the square

Solution;

First, we will find the area of the triangle.

Using heron's formula we get,

s = $\frac{A+B+C}{2}$ where,

s→ semi perimeter of the triangle

s = $\frac{10+17+21}{2}$ = 24cm

Area of triangle ABC= $\sqrt{s(s-A)(s-B)(s-C)}$

Area of triangle ABC = 84$cm^{2}$

from this, we get the height of the triangle = 8 cm

Let 'x' be the side of the square

PQ=QR=RS=SP = x cm

Area of small triangle APQ = x(x-8)/2.....(1)

Area of trapezoid PQCB = x(21 + x)/2.....(2)

Adding (1) and (2) we get the area of triangle ABC,

So, 8$x$ - $x^{2}$ + 21$x$ + $x^{2}$ = 84x2

29x = 168

x = 5.8

Hence, the side of the square is 5.8 cm

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Answer 2

Answer:- The length of the side of the square = 5.8 cm

Given in the question that:-

Let us name the triangle as triangle ABC

Sides of the triangle are given

AB= 21 cm, BC=17 cm and CA=10 cm

We need to find the length of the side of the square. For that, first we need to find the area of Triangle ABC with Heron's formula.

We need to find the Semi-perimeter of the triangle to find area by Heron's formula,

The sides are marked as a, b and c. Such that

a = 21 cm, b = 17 cm and c = 10 cm

Semi-perimeter (s) = Perimeter of triangle/2

                               = (a+b+c)/2

                               = (21+17+10)/2

                               = 48/2

                               = 24 cm

Therefore,

Area of triangle ABC by Heron's formula = √s(s-a)(s-b)(s-c)

                                                                    = √24(24-21)(24-17)(24-10)

                                                                    = √24×3×7×14

                                                                    = √7056

                                                                    = 84 cm²

Now, we need to find the height of the triangle, so

Area of triangle ABC = 1/2×base×height

              84 cm²         = 1/2×21cm×height

       84cm²×2÷21cm  = height

       4×2 cm                = height

                   8 cm        = height

Now, we get the height of the triangle = 8 cm

Now let us assume the side of the square be = x cm

Let us name the square as PQRS

Therefore,

PQ=QR=RS=SP = x cm(all sides of a square are equal)

Now, a small triangle APQ and a trapezium PQCB is formed.

So, area of small triangle APQ = x(x-8)/2                 eq.(1)

and,

Area of trapezium PQCB = x(21 + x)/2                        eq(2)

Now,

By adding equation (1) and (2) we get the area of triangle ABC,

We get,

8x+21x= 84x2

29x = 168

 x    = 168/29

    x = 5.8 cm

Therefore, the side of the square is = 5.8 cm

To know more about the given topic please go through the following

Link1:- https://brainly.in/question/11566151?

Link2:- https://brainly.in/question/6044138?

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