Junior Cycle Mathematics Trigonometry: The diagram below shows Tom's la

The diagram below shows Tom's ladder leaning against a vertical wall - Junior Cycle Mathematics - Question 10 - 2018

Question 10

The diagram below shows Tom's ladder leaning against a vertical wall. The ladder is 5 m long. It makes an angle of $B$ with the horizontal ground. The distance f... show full transcript

Worked Solution & Example Answer:The diagram below shows Tom's ladder leaning against a vertical wall - Junior Cycle Mathematics - Question 10 - 2018

Step 1

Use the theorem of Pythagoras to find the value of $y$

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Answer

According to the Pythagorean theorem, we have:

$3^2 + y^2 = 5^2$

Substituting the known values gives:

$9 + y^2 = 25$

Rearranging the equation:

$y^2 = 25 - 9$

Thus,

$y^2 = 16$

Taking the square root of both sides results in:

$y = \sqrt{16} = 4 ext{ m}$

Step 2

(i) Below are three statements about the angle $B$ in the diagram. Put a tick (✓) in the correct box to show which one is true.

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Answer

Upon evaluating the ratios:

The adjacent side is the distance from the wall: 3 m
The hypotenuse is the length of the ladder: 5 m

Thus, we find:

$\cos B = \frac{3}{5}$

This means the correct statement is:

cos $B = \frac{3}{5}$ (Tick this box)

Step 3

(ii) Hence work out the size of the angle $B$.

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Answer

To find the angle $B$ , we use the cosine inverse:

$B = \cos^{-1}\left(\frac{3}{5}\right)$

Using a calculator, we calculate:

$B \approx 53.1^{\circ}$

Thus, the size of angle $B$ is approximately:

$B = 53^{\circ}$ (to one decimal place)

Step 4

Use similar triangles to find the value of $x$, the vertical height of Jamie's ladder.

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Answer

For the triangles formed by the ladders and the wall, we have:

From Cameron's ladder, the height is 6 m and the distance from the wall is 5.8 m. The ratio is:

$\frac{6}{5.8} = \frac{x}{9}$

Cross-multiplying gives:

$6 \cdot 9 = 5.8 \cdot x$

Calculating gives:

$54 = 5.8x$

Then, solving for $x$ :

$x = \frac{54}{5.8} \approx 9.31 ext{ m}$

Hence the vertical height of Jamie's ladder is:

$x \approx 9.31 ext{ m}$

The diagram below shows Tom's ladder leaning against a vertical wall - Junior Cycle Mathematics - Question 10 - 2018

Worked Solution & Example Answer:The diagram below shows Tom's ladder leaning against a vertical wall - Junior Cycle Mathematics - Question 10 - 2018

Use the theorem of Pythagoras to find the value of $y$

(i) Below are three statements about the angle $B$ in the diagram. Put a tick (✓) in the correct box to show which one is true.

(ii) Hence work out the size of the angle $B$.

Use similar triangles to find the value of $x$, the vertical height of Jamie's ladder.

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