Exercise 15 Page 619

Practice makes perfect

a Let's begin with recalling the Geometric Mean Altitude Theorem.

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

x/h=h/y or h=sqrt(x*y) Now, let's take a look at the given picture.

Using the theorem we recalled at the beginning, we can write an equation. x=sqrt(8*12.5) Now let's solve the equation.

x=sqrt(8*12.5)

Multiply

x=sqrt(100)

CalcRoot

Calculate root

x=10

The value of x is 10.

b Let's begin with recalling the Pythagorean Theorem.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a^2+ b^2= c^2 Now, let's look at the given picture.

Since an altitude in a triangle divides this triangle into two right triangles, we can use the Pythagorean Theorem to evaluate the value of y.

10^2+ 8^2= y^2

CalcPow

Calculate power

100+64=y^2

AddTerms

Add terms

164=y^2

RearrangeEqn

Rearrange equation

y^2=164

Next we will take a square root of both sides of the equation. Notice that since y represents the side length, we will consider only the positive case when taking a square root of y^2.

y^2=164

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(y^2)=sqrt(164)

SqrtPowToNumber

sqrt(a^2)=a

y=sqrt(164)

CalcRoot

Calculate root

y=12.8062...

RoundDec

Round to 1 decimal place(s)

y≈ 12.8

The value of y is approximately 12.8.

c Again, let's begin with recalling the Pythagorean Theorem.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a^2+ b^2= c^2 Next we will look at the given picture for the last time.

Having in mind that an altitude in a triangle divides this triangle into two right triangles, we can again use the Pythagorean Theorem to evaluate the value of z.

10^2+ 12.5^2= z^2

CalcPow

Calculate power

100+156.25=z^2

AddTerms

Add terms

256.25=z^2

RearrangeEqn

Rearrange equation

z^2=256.25

Next we will take a square root of both sides of the equation. Notice that since z represents the side length, we will consider only positive case when taking a square root of z^2.

z^2=256.25

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(z^2)=sqrt(256.25)

SqrtPowToNumber

sqrt(a^2)=a

z=sqrt(256.25)

CalcRoot

Calculate root

z=16.0078...

RoundInt

Round to nearest integer