## Envision Math Grade 8 Volume 1 Chapter 7 Understand The Converse Of The Pythagorean Theorem

**Page 381 Focus On Math Practices Answer**

We have already proved that the sum of the areas of two squares with sides a and b is the same with the area of a square with side c

We have proved this relationship with all right-angle triangles.

Instead of the actual length of the sides, we have used the symbols a, b, and c

while proving the relationship for Kelly’s triangle.

Hence, this relationship is true for any right-angle triangle. The only mandatory condition is a triangle to be a right angle triangle.

Yes, another right-angle triangle drawn by Kelly will also have the same relationship.

This relationship is true for any right-angle triangle.

**Page 382 Essential Question Answer**

Let the right-angle triangle be ABC

The base of the triangle is a

The height of the triangle is b and hypotenuse side is c

According to the Pythagorean theorem, the sum of the square of the base and height is equal to the square of the hypotenuse side.

So, according to the Pythagorean theorem

a^{2}+ b^{2} = c^{2}

According to the Pythagorean theorem, the sum of the square of two sides of a right-angle triangle is equal to the square of the hypotenuse side.

Hence, the relationship between the side lengths of the right angle triangle is given as a^{2}+ b^{2} = c^{2}

**Page 382 Try It Answer**

Given: Sides of a right-angle triangle is 15cm, 25cm and 20cm

To find : The equation that describes the relationship between the sides of the right-angle triangle.

We will use the Pythagorean theorem to find the relationship between the length of the sides of the triangle.

Let, the base (a) of the right angle triangle be 15cm

Let the height(b) of the right angle triangle be 20cm

Let the hypotenuse side (c) by 25cm

According to the Pythagorean theorem, a^{2}+ b^{2} = c^{2}

Substituting the value, we get,

15^{2} + 20^{2} = 25^{2}

We will solve the equation 15^{2} + 20^{2} = 25^{2}, to prove whether the relationship is true.

225 + 400 = 625

Combining common terms, we get,

625 = 625

Hence, the relationship is true.

So, according to the Pythagorean theorem, 15^{2} + 20^{2} = 25^{2}

The equation that describes the relationship between the length of the sides of right triangle is 15^{2} + 20^{2} = 25^{2}

**Page 382 Convince Me Answer**

The diagram is

In the above diagram, instead of the actual lengths of the side, symbols like a, b, and c is used.

Since the length of the sides is not given in actual numbers hence, this formula can be applied to any right-angle triangle.

The only mandatory condition is that triangle has to be the right triangle.

Hence, the Pythagorean theorem can be applied to the all right triangle.

The side opposite to the greatest angle is the longest side of the right-angle triangle.

The Pythagorean theorem is applied to all right triangles because the lengths of the sides are given in symbol rather than the actual lengths.

**Page 383 Try It Answer**

Given:

The hypotenuse = 32 meters.

One leg = 18 meters.

To find : The length of the other leg.

We will use the Pythagorean theorem to find the length of the other leg.

Since we are not given the length of which side is 18meters

We will consider a = 18meters and solve for b

The hypotenuse of the triangle is c = 32 meters

So, the length of the other leg is 10√7

The length of the another leg is 10√7

**Page 384 Exercise 1 Answer**

Let the right-angle triangle be ABC

The base of the triangle is a

The height of the triangle is b and the hypotenuse side of the triangle is c

According to the Pythagorean theorem, the sum of the square of the base and height is equal to the square of the hypotenuse side.

So, according to the Pythagorean theorem a^{2}+ b^{2} = c^{2}

According to the Pythagorean theorem, the sum of the square of two sides of a right-angle triangle is equal to the square of the hypotenuse side.

Hence, the relationship between the side lengths of the right angle triangle is given as a^{2}+ b^{2} = c^{2}

**Page 384 Exercise 2 Answer**

The given diagram is

The diagram shows that the lengths of the legs are 4unit and 3 unit

While the length of the hypotenuse side is 5unit

we can see that the length of the hypotenuse is longer than other sides.

For any right triangle, the longest side is the hypotenuse side and the Pythagorean theorem is applied to only the right triangle.

Hence, the requested condition is that the square that would form the side of the hypotenuse would have the longest side.

Another condition is that each side of the triangle must be smaller than the sum of the other two sides.

If the side of the triangle are a, b, and c

Then a < b + c, b < a + c and c < a + b

No, any three squares cannot form the right triangle. The square forming the hypotenuse side would have the longest side and another condition is that each side of the triangle should be smaller than the sum of the other sides.

**Page 384 Exercise 3 Answer**

Given:

To find : Whether Xavier has given the correct length of the hypotenuse side.

In a right-angled triangle, 90-degree angle is the largest angle. The side opposite to the largest angle in a triangle is the longest side.

The triangle is

The length of the two legs are 21 units and 28 units

According to Xavier, the length of the hypotenuse is 18.5 units

But since the length of the other sides are 21 units and 28 units which is longer than Xavier’s hypotenuse side.

But since the length of the hypotenuse cannot be shorter than the other two sides, hence, the length given by Xavier is incorrect.

The length of the hypotenuse side should be longer than the other two sides. But the length of the hypotenuse side given by Xavier is less than the other two sides, hence, the length given by the hypotenuse side is an incorrect length.

**Page 384 Exercise 4 Answer**

Given: sides of a triangle is 4 and 5

To find: hypotenuse of triangle

We will put the given values in

Hypotenuse = \(\sqrt{\text { base }^2+\text { perpendicular }{ }^2}\)

Hypotenuse of the triangle is ≈6

**Page 384 Exercise 5 Answer**

Given: perpendicular = 8 and hypotenuse = 14

To find: base of the triangle

We will use the Pythagoras formula and find the dimension of side.

Putting all the given values in Pythagoras formula

Base of the triangle is approximately 12 ft

**Page 384 Exercise 6 Answer**

Given: perpendicular = 3.7mm and base = 7.5 mm

To find: hypotenuse of the triangle

We will use the Pythagoras formula and find the dimension of side.

Putting all the given values in Pythagoras formula

Hypotenuse of the given triangle is approximately 8

**Page 385 Exercise 9 Answer**

Given: perpendicular = 4x + 4 and base = 3x where x = 15

To find: hypotenuse of the triangle

We will use the Pythagoras formula and find the dimension of side.

Putting all the given values in Pythagoras formula and solving as

Hypotenuse of the given triangle is approximately 78 units.

**Page 385 Exercise 10 Answer**

Given: perpendicular=12.9cm and hypotenuse = 15.3 cm

To find: base of the triangle

We will use the Pythagoras formula and find the dimension of side.

Putting all the given values in Pythagoras formula and solving as

Base of the given triangle is approximately a = ≈ 8 cm

**Page 385 Exercise 11 Answer**

Given: perpendicular=10m and base = 24m

To find: hypotenuse of the triangle

We will use the Pythagoras formula and find the dimension of side.

Putting all the given values in Pythagoras formula and solving as

Hypotenuse of the given triangle is approximately 26 m

**Page 385 Exercise 12 Answer**

Given: base = 2ft and hypotenuse = 9 ft

To find: perpendicular of the triangle

We will use the Pythagoras formula and find the dimension of side.

Putting all the given values in Pythagoras formula and solving as

Perpendicular of the given triangle is approximately 8

**Page 386 Exercise 13 Answer**

Given: A triangle where two of the legs are 32 cm and 26 cm

To find: Hypotenuse of triangle

We will use the Pythagoras formula and find the dimension of the required side.

Putting all the given values in Pythagoras formula and solving as

Length of hypotenuse of the given triangle is approximately 41 cm

Given: A triangle where two of the legs are 32 cm and 26 cm

To find: What mistake might the student have made?

We will write the dimension of hypotenuse calculated and match it with given one.

As per the calculation using

(hypotenuse)^{2} = base^{2} + perpendicular^{2}

Hypotenuse is approximately 41 cm.

The mistake might be done in taking the values of legs incorrectly or in the formula.

Mistake might be in the formula taken or dimensions taken.

**Page 386 Exercise 14 Answer**

Given: A figure of triangle where base is 12.75 and hypotenuse is 37.25

To find: unknown side of the triangle

We will use the Pythagoras formula and find the dimension of the required side.

Putting all the given values in Pythagoras formula and solving as

Length of perpendicular of the triangle is 35

**Page 386 Exercise 16 Answer**

Given: A figure of triangle where two sides are 36 and 15 ft

To find: unknown side of the triangle

We will use the Pythagoras formula and find the dimension of the required side.

Putting all the given values in Pythagoras formula and solving as

Length of hypotenuse of the given triangle is 39 ft.

**Page 386 Exercise 17 Answer**

Given: A figure of triangle where base is 11.25 cm and hypotenuse is 35.25 cm

To find: unknown side of the triangle

We will use the Pythagoras formula and find the dimension of the required side.

Putting all the given values in Pythagoras formula and solving as

Length of perpendicular of the given triangle is approximately33 cm.