Bokhairaidn
Answered

Uncover valuable information and solutions with IDNLearn.com's extensive Q&A platform. Ask anything and receive thorough, reliable answers from our community of experienced professionals.

The diagram shows a 9cm x 7cm rectangle-based pyramid. all the diagonal sides - TA, TB, TC, AND TD- are length 12cm. M is the midpoint of the rectangular base.
Work out angle TAC, to 1 decimal place.

The Diagram Shows A 9cm X 7cm Rectanglebased Pyramid All The Diagonal Sides TA TB TC AND TD Are Length 12cm M Is The Midpoint Of The Rectangular Base Work Out A class=

Sagot :

Answer:

∠TAC is approximately equal to 61.6°

Step-by-step explanation:

The given parameters for the pyramid are;

The dimension for the rectangular base are; Length = 9 cm, width = 7 cm

The length of the diagonal sides, TA, TB, TC, and TD = 12 cm each

The midpoint of the rectangular base = Point M

The diagonal AC = AM + MC

AM = MC as given M is the midpoint of the rectangular base

∴ AC = AM + MC = 2·AM

By Pythagoras' theorem, AC = √(9² + 7²) = √130

AC = √130 cm

∴ AM = AC/2 = (√130)/2 cm

Alternatively, AM = √((9/2 cm)² + (7/2 cm)²) = √(32.5) cm

∠TAC = ∠TAM

By trigonometric ratios, we have;

[tex]cos (\theta) = \dfrac{Length \ of \ adjacent \ side \ to \ angle }{Length \ of \ hypotenuse\ side \ to \ angle}[/tex]

[tex]\therefore cos (\angle TAM) = cos (\angle TAC) = \dfrac{\left (\dfrac{\sqrt{130} }{2} \right )}{12} = \dfrac{\sqrt{130} }{2 \times 12} = \dfrac{\sqrt{130} }{24}[/tex]

[tex]\angle TAC = arccos \left ( \dfrac{\sqrt{130} }{24} \right ) \approx 61.6 ^{\circ} \ to 1 \ decimal \ place[/tex]

MrRoyalidn

The measure of angle TAC is 61.6 degrees

The given parameters are:

[tex]\mathbf{AB = 7}[/tex]

[tex]\mathbf{BC = 9}[/tex]

Start by calculating length AC, using the following Pythagoras theorem.

[tex]\mathbf{AC = \sqrt{AB^2 + BC^2}}[/tex]

So, we have:

[tex]\mathbf{AC = \sqrt{7^2 + 9^2}}[/tex]

[tex]\mathbf{AC = \sqrt{49 + 81}}[/tex]

[tex]\mathbf{AC = \sqrt{130}}[/tex]

[tex]\mathbf{AC = 11.40}[/tex]

Next, we calculate side length AM

[tex]\mathbf{AM = \frac{AC}{2}}[/tex]

So, we have:

[tex]\mathbf{AM = \frac{11.40}{2}}[/tex]

[tex]\mathbf{AM = 5.70}[/tex]

The measure of angle TAC is calculated using the following cosine ratio

[tex]\mathbf{cos(A)= \frac{AM}{TA}}[/tex]

So, we have:

[tex]\mathbf{cos(A)= \frac{5.70}{12}}[/tex]

[tex]\mathbf{cos(A)= 0.475}[/tex]

Take arc cos of both sides

[tex]\mathbf{A= 61.6}[/tex]

Hence, the measure of angle TAC is 61.6 degrees

Read more about pyramids at:

https://brainly.com/question/15591430

We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.