Drawback 1 :

One of many diagonals of a rectangle is 20 cm lengthy. If the distinction between its size and width is 4 cm, then discover the world of the rectangle.

Drawback 2 :

If the peak of a triangle is 17 inches lower than the size of its base and the size of the hypotenuse is 25 inches, discover the bottom and the peak.

Drawback 3 :

The perimeters of an equilateral triangle are shortened by 12 items, 13 items, 14 items respectively and a proper angle triangle is shaped. Discover the size of facet of the equilateral triangle.

Drawback 4 :

The bottom of a proper triangle is 32 cm larger than its peak and the hypotenuse is 7 cm larger than its base. Discover the lengths of the three sides of the precise triangle.

Drawback 5 :

From a practice station, one practice heads north, and one other heads east. A while later, the northbound practice has traveled 12 miles, and the eastbound practice has traveled 16 miles. How far aside are the 2 trains, measured in a straight line ?

Drawback 6 :

The three sides of a proper triangle are within the ratio 3 : 4 : 5 and the world of the triangle is 24 sq. cm. Discover the perimeters of the triangle.

## Solutions

1. Reply :

Let x be the size of the rectangle.

As a result of the distinction between its size and width is 4 cm, its width have to be both (x + 4) cm or (x – 4) cm.

Let’s take the width as (x + 4) cm.

Draw a sketch.

In the precise triangle, in line with Pythagorean theorem, we now have

(x + 4)^{2 }+ x^{2 }= 20^{2}

Simplify.

x^{2 }+ 2(x)(4) + 4^{2} + x^{2 }= 400

x^{2 }+ 8x + 16 + x^{2 }= 400

2x^{2 }+ 8x + 16^{ }= 400

Subtract 400 from both sides.

2x^{2 }+ 8x – 384^{ }= 0

Divide both sides by 2.

x^{2 }+ 4x – 192^{ }= 0

(x + 16)(x – 12) = 0

x + 16 = 0 or x – 12 = 0

x = -16 or x = 12

As a result of the size of a rectangle can by no means be a adverse worth, we are able to ignore x = -16.

So, the worth of x is 12.

Then,

x + 4 = 12 + 4

x + 4 = 16

Subsequently,

size = 12 cm

width = 16 cm

Space of the triangle :

= l ⋅ w

= 12 ⋅ 16

= 192 cm^{2}

So, the world of the rectangle is 192 sq. cm.

If the peak of a triangle is 17 inches lower than the size of its base and the size of the hypotenuse is 25 inches, discover the bottom and the peak.

2. Reply :

Draw a sketch.

In the precise triangle above, in line with Pythagorean theorem, we now have

(x – 17)^{2 }+ x^{2 }= 25^{2}

Simplify.

x^{2 }– 2(x)(17) + 17^{2} + x^{2 }= 625

x^{2 }– 34x + 289 + x^{2 }= 625

2x^{2 }– 34x + 289^{ }= 625

Subtract 625 from both sides.

2x^{2 }– 34x – 336^{ }= 0

Divide both sides by 2.

x^{2 }– 17x – 168^{ }= 0

(x + 7)(x – 24) = 0

x + 7 = 0 or x – 24 = 0

x = -7 or x = 24

As a result of the bottom of a triangle can by no means be a adverse worth, we are able to ignore x = -7.

So, the worth x is 24.

Then,

x – 17 = 24 – 17

x – 17 = 7

So, the bottom and peak of the precise triangle are 24 inches and seven inches respectively.

3. Reply :

Let x be the size of facet of the equilateral facet.

Given : The perimeters of an equilateral triangle are shortened by 12 items, 13 items, 14 items and a proper angle triangle is shaped.

Then, new lengths of the triangle are

(x – 12), (x – 13) and (x – 14)

These are the perimeters of a proper angle triangle and (x – 12) is the longest facet.

Draw a sketch.

As a result of the longest facet of the precise triangle is hypotenuse, (x – 12) will be thought-about to be the size of the hypotenuse.

In keeping with Pythagorean theorem, the sq. of the hypotenuse is the same as sum of the squares of different two sides.

Then, we now have ,

(x – 13)^{2} + (x – 14)^{2} = (x – 12)^{2}

x^{2} – 2(x)(13) + 13^{2} + x^{2} – 2(x)(14) + 14^{2} = x^{2} – 2(x)(12) + 144

x^{2} – 26x + 169 + x^{2} – 28x + 196 = x^{2} – 24x + 144

2x^{2} – 54x + 365 = x^{2} – 24x + 144

x^{2} – 30x + 221 = 0

(x – 13)(x – 17) = 0

x = 13 or x = 17

If we take x = 13, the perimeters of the precise triangle are

x – 12 = 1, x – 13 = 0, x – 14 = -1

When x = 13, we get one of many sides is zero and the signal of the one other facet is adverse.

Then, x = 13 cannot be accepted.

If we take x = 17, the perimeters of the precise triangle are

x – 12 = 5, x – 13 = 4, x – 14 = 3

All the three sides of the precise angle are constructive when x = 17.

Furthermore, the lengths 5, 4 and three fulfill the Pythagorean theorem.

That’s,

5^{2} = 4^{2} + 3^{2}

25 = 16 + 9

25 = 25

Subsequently, x = 17 will be accepted.

So, the size of facet of equilateral triangle is 17 items.

4. Reply :

Let x be the peak of the precise triangle.

Then, we now have

Base = x + 32

Hypotenuse = (x + 32) + 8 = x + 40

Draw a sketch.

In the precise triangle above, in line with Pythagorean theorem, we now have

x^{2} + (x + 31)^{2}^{ }= (x + 32)^{2}

Simplify.

x^{2 }+ x^{2} + 2(x)(31) + 31^{2 }^{ }= x^{2 }+ 2(x)(32) + 32^{2}

2x^{2 }+ 62x + 961 = x^{2} + 64x + 1024

x^{2 }– 2x – 63 = 0

Issue.

(x + 7)(x – 9) = 0

x + 7 = 0 or x – 9 = 0

x = -7 or x = 9

As a result of the peak of a triangle can by no means be a adverse worth, we are able to ignore x = -7.

So, the worth x is 9.

Then,

x + 31 = 9 + 31 = 40

x + 32 = 9 + 32 = 41

So, the lengths of the three sides of the precise triangle are 9 cm, 40 cm and 41 cm.

5. Reply :

Let the 2 trains be x miles aside.

Draw a sketch.

In the precise triangle above, in line with Pythagorean theorem, we now have

x^{2 }= 12^{2} + 16^{2}

Simplify and clear up for x.

x^{2 }= 144 + 256

x^{2 }= 400

x^{2 }= 20^{2}

x = 20

So, the 2 trains are 20 miles aside.

6. Reply :

From the given ratio 3 : 4 : 5, the perimeters of the triangle will be assumed as

3x, 4x, 5x

At all times the longest facet of the precise triangle is hypotenuse.

Then, the size of the hypotenuse is 5x.

Draw a sketch.

In the precise triangle above, the bottom is 3x and the peak is 4x.

Given : Space of the triangle is 24 sq. cm.

Space = 24 cm^{2}

Write the method for space of triangle.

1/2 ⋅ base ⋅ peak = 24

Substitute 3x for base and 4x for peak.

1/2 ⋅ 3x ⋅ 4x = 24

6x^{2} = 24

Divide both sides by 6.

x^{2} = 4

x^{2} = 2^{2}

x = 2

Then, we now have

3x = 3(2) = 6

4x = 4(2) = 8

5x = 5(2) = 10

So, the three sides of the triangle are 6 cm, 8 cm and 10 cm.

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