Practice Problems (Junior Cert Mathematics): Revision Notes

Practice Problems

Problems:

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Problem 1:

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Problem 2:

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Problem 3:

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Problem 4:

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Problem 5:

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Problem 6:

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Solutions:

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Problem 1:

Step 1: Identify the sides of the triangle.

• The given sides are the legs of the triangle.
• The hypotenuse is the unknown side, labelled as $c$.

Step 2: Write down the Pythagoras' Theorem formula. $c^2 = a^2 + b^2$ This formula states that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

Step 3: Substitute the known values into the formula. $c^2 = 3^2 + 4^2$

Step 4: Calculate the squares of the sides. $c^2 = 9 + 16$

Step 5: Add the squares. $c^2 = 25$

Step 6: Find $c$ by taking the square root of both sides. $c = \sqrt{25} = 5 \, \text{cm}$

Explanation: The square root of $25$ gives the length of the hypotenuse. Since the hypotenuse is always the longest side, the answer makes sense.

Final Answer: The length of the hypotenuse is $5$ $cm$.

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Problem 2:

Step 1: Identify the sides of the triangle.

• The hypotenuse is $10$ $cm$.
• One leg is $6$ $cm$.
• The unknown side is labelled as $b$.

Step 2: Write down the Pythagoras' Theorem formula. $c^2 = a^2 + b^2$ Since $c$ is the hypotenuse, rearrange the formula to solve for $b$: $b^2 = c^2 - a^2$

Step 3: Substitute the known values into the formula. $b^2 = 10^2 - 6^2$

Step 4: Calculate the squares of the sides. $b^2 = 100 - 36$

Step 5: Subtract the squares. $b^2 = 64$

Step 6: Find $b$ by taking the square root of both sides. $b = \sqrt{64} = 8 \, \text{cm}$

Explanation: Taking the square root of $64$ gives the length of the unknown side. The answer fits within the context, as the hypotenuse should be the longest side.

Final Answer: The length of the other side is $8$ $cm$.

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Problem 3:

Step 1: Identify the sides of the triangle.

• The ladder forms the hypotenuse, which is $13$ $meters$.
• The distance from the wall to the base of the ladder is one side of the triangle, which is $5$ $meters$.
• The height up the wall (unknown) is labelled as $h$.

Step 2: Write down the Pythagoras' Theorem formula. $c^2 = a^2 + b^2$ Since $c$ is the hypotenuse, rearrange the formula to solve for $h$ (height): $h^2 = c^2 - b^2$

Step 3: Substitute the known values into the formula. $h^2 = 13^2 - 5^2$

Step 4: Calculate the squares of the sides. $h^2 = 169 - 25$

Step 5: Subtract the squares. $h^2 = 144$

Step 6: Find $h$ by taking the square root of both sides. $h = \sqrt{144} = 12 \, \text{meters}$

Explanation: The square root of $144$ gives the height the ladder reaches on the wall. The solution is reasonable, as the ladder's height is less than the ladder's length.

Final Answer: The ladder reaches $12$ $meters$ up the wall.

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Problem 4:

Step 1: Identify the sides of the triangle.

• Angle A is $30°$.
• The hypotenuse is $10$ $cm$.
• The side opposite $Angle$ $A$ is unknown $(x)$.

Step 2: Choose the correct trigonometric ratio. Since the opposite side and hypotenuse are involved, use the Sine ratio: $\sin(A) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

Step 3: Set up the equation. $\sin(30°) = \frac{x}{10}$

Step 4: Solve for $x$. First, find $\sin(30°)$ using a calculator: $\sin(30°) = 0.5$ Now, substitute this value into the equation: $0.5 = \frac{x}{10}$ Multiply both sides by $10$: $x = 0.5 \times 10 = 5 \, \text{cm}$

Explanation: Using the sine function helps find the opposite side when the hypotenuse and angle are known. Multiplying by the hypotenuse gives the correct side length.

Final Answer: The length of the side opposite Angle A is $5$ $cm$.

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Problem 5:

Step 1: Identify the sides of the triangle.

• The adjacent side is $8$ $cm$.
• The hypotenuse is $10$ $cm$.
• Angle A is unknown.

Step 2: Choose the correct trigonometric ratio. Since the adjacent side and hypotenuse are involved, use the Cosine ratio: $\cos(A) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

Step 3: Set up the equation. $\cos(A) = \frac{8}{10}$

Step 4: Simplify and solve for $A$. First, calculate the fraction: $\cos(A) = 0.8$ Now, find $Angle$ $A$ using the inverse cosine function: $A = \cos^{-1}(0.8)$ Using a calculator: $A \approx 36.87°$

Explanation: The inverse cosine function helps find the angle when the sides are known. The calculator's inverse function is used to work backwards from the ratio to the angle.

Final Answer: $Angle$ $A$ is approximately $36.9°$.

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Problem 6:

Step 1: Identify the sides of the triangle.

• Angle A is $40°$.
• The adjacent side is $7$ $cm$.
• The opposite side is unknown $(x)$.

Step 2: Choose the correct trigonometric ratio. Since the opposite and adjacent sides are involved, use the $Tangent$ ratio: $\tan(A) = \frac{\text{Opposite}}{\text{Adjacent}}$

Step 3: Set up the equation. $\tan(40°) = \frac{x}{7}$

Step 4: Solve for $x$. First, find $\tan(40°)$ using a calculator: $\tan(40°) \approx 0.8391$ Now, substitute this value into the equation: $0.8391 = \frac{x}{7}$ Multiply both sides by $7$: $x = 0.8391 \times 7 = 5.87 \, \text{cm}$

Explanation: Using the tangent function helps find the opposite side when the adjacent side and angle are known. Multiplying by the adjacent side gives the correct side length.

Final Answer: The length of the side opposite $Angle$ $A$ is $5.87$ $cm$.

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