High School

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Triangle ABC has sides AB= 17cm, AC= 13cm and BC= 23cm, as shown

below.

13 cm

23cm

17 cm

Diagram not drawn to scale

Calculate the size of angle CAB to the nearest integer.

(3 marks)

Triangle ABC has sides AB 17cm AC 13cm and BC 23cm as shown below 13 cm 23cm 17 cm Diagram not drawn to scale Calculate

Answer :

Final answer:

The measure of angle CAB is approximately 108°.

Explanation:

To find the measure of angle CAB, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and opposite angles A, B, and C respectively, the following equation holds: c^2 = a^2 + b^2 - 2ab * cos(C).

In this case, we know that side AB is 17cm, side AC is 13cm, and side BC is 23cm. Let's substitute these values into the equation:

23^2 = 17^2 + 13^2 - 2 * 17 * 13 * cos(C)

529 = 289 + 169 - 442 * cos(C)

442 * cos(C) = -71

cos(C) = -71/442

C = arccos(-71/442)

C ≈ 108° (to the nearest integer).

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Rewritten by : Barada

Answer:

[tex]\boxed{m \angle CAB =99^{\circ}}[/tex]

Step-by-step explanation:

We can use the law of cosines to determine measure of ∠CAB

If a, b and c are the three sides of a triangle and C is the angle opposite side c then the law of cosines says
c² = a² + b² - 2ab cos(C)

Here we are asked to find m∠CAB

The side opposite ∠CAB is BC which is 23 cm long. This is c in the formula

The other two sides, a and b are 13 cm and 17 cm

Applying the formula, using C to represent m∠CAB

[tex]23^2=\:13^2+\:17^2-\:2\:\cdot 13\cdot 17\cdot \cos \left(C\right)[/tex]

First switch sides:
[tex]13^2+17^2-2\cdot \:13\cdot \:17\cos \left(C\right)=23^2[/tex]

[tex]\rightarrow 69+289-442\cos \left(C\right)=529\\\\\rightarrow 458 -4 42\cos \left(C\right) = 529\\[/tex]

[tex]\rightarrow -442\cos \left(C\right) = 529-458\\\\\rightarrow -442\cos \left(C\right) = 71\\\\[/tex]

[tex]\rightarrow \cos \left(C\right)=-\dfrac{71}{442}\\\\\rightarrow \cos \left(C\right) = - 0.16063\\\\[/tex]

[tex]\rightarrow C = \cos^{-1} (-0.16063)[/tex]

[tex]\rightarrow C = 99.2437^{\circ}[/tex]

Rounded to the nearest integer, this would become

[tex]\rightarrow C = \boxed{99^{\circ}}[/tex]