A ladder leaning against a wall makes an angle of $60^{\circ}$ with the ground. If the foot of the ladder is 10 m away from the wall, what is the length of the ladder?

The correct answer is Option C: 20 m. Use the 30-60-90 triangle ratio: sides are $x, x\sqrt{3}, 2x$ opposite to $30^\circ, 60^\circ, 90^\circ$ respectively. Here, the side adjacent to $60^\circ$ is $x = 10$, so the hypotenuse is $2x = 20$. Step-by-Step Solution & Concepts: Shortcut Trick: Use the 30-60-90 triangle ratio: sides are $x, x\sqrt{3}, 2x$ opposite to $30^\circ, 60^\circ, 90^\circ$ respectively. Here, the side adjacent to $60^\circ$ is $x = 10$, so the hypotenuse is $2x = 20$. Formula Used: $$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$ Given: Angle of inclination of ladder with ground is 60 degrees, distance of the foot of the ladder from the wall is 10 m. Steps: Identify the triangle configuration: The ladder forms a right-angled triangle with the wall and the ground, where the ladder is the hypotenuse.. Select the appropriate trigonometric ratio: Since we have the adjacent side and need the hypotenuse, we use the cosine ratio.. Calculate the length: Knowing that $\cos(60^\circ) = 0.5$ or $1/2$, we solve for $L$. Answer: C is correct because using the cosine ratio for a $60^\circ$ angle, the length of the ladder is calculated as $10 / \cos(60^\circ) = 20$ m. Exam Tip: This concept is fundamental for vector resolution in Physics and finding components of forces on an inclined plane.

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A ladder leaning against a wall makes an angle of $60°$ with the ground. If the foot of the ladder is 10 m away from the wall, what is the length of the ladder?

10 m

15 m

20 m

10 $\sqrt{3} m$

Correct Answer

📐 MA • Math Trigonometric Ratio MethodMathematicsTrigonometry

Option C

20 m

Quick Summary:

Use the 30-60-90 triangle ratio: sides are $x, x\sqrt{3}, 2x$ opposite to $30°, 60°, 90°$ respectively. Here, the side adjacent to $60°$ is $x = 10$ , so the hypotenuse is $2x = 20$ .

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📐MAMath SolutionTrigonometric Ratio Method

📋 Given

Angle of inclination of ladder with ground is 60 degrees, distance of the foot of the ladder from the wall is 10 m.

🔢 Formula Used

cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}cos(θ)=HypotenuseAdjacent​

⚡ Exam Hall Shortcut / Speed Trick

Use the 30-60-90 triangle ratio: sides are $x, x\sqrt{3}, 2x$ opposite to $30°, 60°, 90°$ respectively. Here, the side adjacent to $60°$ is $x = 10$ , so the hypotenuse is $2x = 20$ .

⚠️ Common Student Trap / Pitfall

Confusing the adjacent side with the opposite side or misidentifying the hypotenuse as the wall height instead of the ladder length.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify the triangle configuration

The ladder forms a right-angled triangle with the wall and the ground, where the ladder is the hypotenuse.

Hypotenuse=L,Adjacent side=10 m,θ=60°\text{Hypotenuse} = L, \text{Adjacent side} = 10\text{ m}, \theta = 60°Hypotenuse=L,Adjacent side=10 m,θ=60°

Select the appropriate trigonometric ratio

Since we have the adjacent side and need the hypotenuse, we use the cosine ratio.

cos⁡(60°)=10L\cos(60°) = \frac{10}{L}cos(60°)=L10​

Calculate the length

Knowing that $\cos(60°) = 0.5$ or $1/2$ , we solve for $L$ .

12=10L  ⟹  L=10×2=20 m\frac{1}{2} = \frac{10}{L} \implies L = 10 \times 2 = 20\text{ m}21​=L10​⟹L=10×2=20 m

✅

C is correct because using the cosine ratio for a $60°$ angle, the length of the ladder is calculated as $10 / \cos(60°) = 20$ m.

Core Concepts Used

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Trigonometric ratios Right-angled triangles 30-60-90 triangle properties

💡 EXAM TIP

This concept is fundamental for vector resolution in Physics and finding components of forces on an inclined plane.

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