Chapter 9.3, Problem 3PSA

a.

To determine

To calculate: Thelength of CD

Answer to Problem 3PSA

The length of CD is $6$

Explanation of Solution

Given:

The measures of the given lengths are as follows:

AD = $4$

BD = $9$

Concept used:

Here, we are using the concept hypotenuse leg theorem.

Calculation:

According to property associated with altitudes in right angled triangle. The altitude to hypotenuse divides the triangles into two similar triangles.

Hence, $\Delta ADC\sim \Delta CDB$

So, $\frac{CD}{AD}=\frac{BD}{CD}$

  $C{D}^2=BD\times AD$

  →$C{D}^2=9\times 4$

  →$CD=\sqrt{36}$

  →$CD=6$

Conclusion: Hence, the length of CD is $6$

b.

To determine

To calculate: Thelength of AC

Answer to Problem 3PSA

The length of AC is $8$

Explanation of Solution

Given:

The measures of given lengths are as follows:

AD = $4$

AB = $16$

Formula used:

Here, we are using Pythagoras theorem

Calculation: According to property associated with altitudes in right angled triangle. The altitude to hypotenuse divides the triangles into two similar triangles.

Hence, $\Delta ADC\sim \Delta CDB$

So, $\frac{CD}{AD}=\frac{BD}{CD}$

Here, AD = $4$ and BD = $12$

So, $C{D}^2=BD\times AD$

  $C{D}^2=48$

And, ${\left(Hypotenuse\right)}^2={\left(Perpendicular\right)}^2+{\left(Base\right)}^2$

  ${\left(AC\right)}^2={\left(CD\right)}^2+{\left(AD\right)}^2$

  ${\left(AC\right)}^2=48+16$

  $AC=\sqrt{64}$

  $AC=8$

Conclusion: Hence, the length of AC is $8$

c.

To determine

To calculate: Thelength of BC

Answer to Problem 3PSA

The length of BC is $4\sqrt{3}$

Explanation of Solution

Given:

The measures of the given lengths are as follows:

BD = $6$

AB = $8$

Concept used:

Here, we are using the concept hypotenuse leg theorem.

Calculation:

According to property associated with altitudes in right angled triangle. The altitude to hypotenuse divides the triangles into two similar triangles.

Hence, $\Delta ADC\sim \Delta CDB$

So, $\frac{CD}{AD}=\frac{BD}{CD}$

  $C{D}^2=BD\times AD$

  $C{D}^2=12$

And, $A{C}^2=12+4$

  $AC=4$ and $AB=8$

  $A{B}^2=A{C}^2+B{C}^2$

  $64=16+B{C}^2$

  $BC=4\sqrt{3}$

Conclusion: Hence, the length of BC is $4\sqrt{3}$

d.

To determine

To calculate: Thelength of AD

Answer to Problem 3PSA

The length of ADis $4$

Explanation of Solution

Given:

The measures of the given lengths are as follows:

CD = $8$

BD = $16$

Concept used:

Here, we are using the concept hypotenuse leg theorem.

Calculation:

According to property associated with altitudes in right angled triangle. The altitude to hypotenuse divides the triangles into two similar triangles.

Hence, $\Delta ADC\sim \Delta CDB$

So, $\frac{CD}{AD}=\frac{BD}{CD}$

  $C{D}^2=BD\times AD$

  $64=16\times AD$

  $AD=4$

Conclusion: Hence, the length of AD is $4$

e.

To determine

To calculate: Thelength of AC

Answer to Problem 3PSA

The length of AC is $9$

Explanation of Solution

Given: The measures of the given lengths are as follows:

AD = $3$

BD = $24$

Concept used:

Here, we are using the concept hypotenuse leg theorem.

Calculation:

According to property associated with altitudes in right angled triangle. The altitude to hypotenuse divides the triangles into two similar triangles.

Hence, $\Delta ADC\sim \Delta CDB$

So, $\frac{CD}{AD}=\frac{BD}{CD}$

  $C{D}^2=BD\times AD$

  ${\left(CD\right)}^2=3\times 24$

  ${\left(CD\right)}^2=72$

  ${\left(AC\right)}^2={\left(CD\right)}^2+{\left(AD\right)}^2$

  ${\left(AC\right)}^2=72+9$

  $AC=9$

Conclusion: Hence, the length of AC is $9$

f.

To determine

To calculate: Thelength of AB

Explanation of Solution

Given:

The measures of the given lengths are as follows:

BC = $8$

BD = $20$

Concept used:

Here, we are using the Pythagoras theorem.

Calculation:

Here, ${\left(BC\right)}^2={\left(BD\right)}^2+{\left(CD\right)}^2$

  $64-400=CD$

Here, the value of CD will be negative which is not possible.