Chapter 9.4, Problem 12PSB

a.

To determine

To calculate: Thelength of CD

Answer to Problem 12PSB

The length of CD is $\sqrt{28}$

Explanation of Solution

Given:

  $\angle ACB$ is a right angle

  $\overline{CD}\perp \overline{AB}$

Lengths of AD and BD are $7$ and $4$

Concept used:

Here, we are using hypotenuse leg similarity rule.

Calculation:

In $\Delta ACD$ and $\Delta DCB$

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

  $\therefore \Delta ACD\sim \Delta DCB$

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

  →$C{D}^2=BD\times AD$

  $C{D}^2=4\times 7$

Conclusion: Hence, the length of CD is $\sqrt{28}$

  $CD=\sqrt{28}$

b.

To determine

To calculate: Thelength of CB

Answer to Problem 12PSB

The length of CB is $10$

Explanation of Solution

Given:

  $\angle ACB$ is a right angle

  $\overline{CD}\perp \overline{AB}$

Lengths of CD and DB are $8$ and $6$

Concept used:

Here, we are using Pythagoras theorem.

  ${\left(Hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

Calculation:

Here, the formula is

  ${\left(Hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

  $C{B}^2=C{D}^2+D{B}^2$

  $C{B}^2=64+36$

  $CB=10$

Conclusion: Hence, the length of CB is $10$

c.

To determine

To calculate: The length of AB

Answer to Problem 12PSB

The length of AB is $32$

Explanation of Solution

Given:

  $\angle ACB$ is a right angle

  $\overline{CD}\perp \overline{AB}$

Lengths of BC and BD are $8$ and $2$

Concept used:

Here, we are using Pythagoras theorem.

  ${\left(Hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

Calculation:

Here, the formula is

  ${\left(Hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

  $B{C}^2=B{D}^2+C{D}^2$

  $8^2=2^2+C{D}^2$

  $64=4+C{D}^2$

  $C{D}^2=60$

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

  $60=AD\times 2$

  $AD=30$

  $AB=AD+DB$

  →$30+2=32$

Conclusion: Hence, the length of AB is $32$

d.

To determine

The length of CB

Answer to Problem 12PSB

The length of CB is $20$

Explanation of Solution

Given:

  $\angle ACB$ is a right angle

  $\overline{CD}\perp \overline{AB}$

Lengths of AC and AB are $21$ and $29$

Concept used:

Here, we are using Pythagoras theorem.

  ${\left(Hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

Calculation:

Here, the formula is

  ${\left(Hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

  $A{B}^2=A{C}^2+C{B}^2$

  $841=441+C{B}^2$

  $400=C{B}^2$

  $CB=20$

Conclusion: Hence, the length of CB is $20$