JEE Main 2019 (10 January Shift 1)

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Animated Solution for Mathematics - Trigonometry: Consider a triangular plot ABC with sides AB=7m, BC=5m and CA=6m. A vertical lamp-post at the mid point D of AC subtends an angle $3 0^{\circ}$ at B. The height (in m) of the lamp-post is:

Q: Why can't we just use the Sine Rule to find $BD$?

You absolutely could! But it would involve finding $\cos(\angle C)$ or $\angle A$ first using the Cosine Rule, then finding the angle $\angle B$, and then applying the Sine Rule. It is a much longer, more winding path. In the JEE, time is currency. The Apollonius Theorem is the 'express lane' for medians. Always look for the most direct geometric property.

Q: How do I know $D$ is the midpoint?

The problem explicitly states $D$ is the midpoint of $AC$. Always read the problem statement carefully—sometimes the most crucial piece of information is hidden in a simple sentence. If $D$ were not the midpoint, $BD$ would just be a cevian, and we would need Stewart's Theorem instead of Apollonius.

Select Answer:

Visualized Solution

Visualizing the Triangular Plot $A B C$

Given sides of $△ A B C$ :
$A B = 7$
$B C = 5$
$C A = 6$

Locating the Lamp-post at Midpoint $D$

$D$ is the midpoint of $A C$ .
So, $A D = D C = 3$ .
A vertical lamp-post $D P$ of height $h$ stands at $D$ .

The Angle Subtended at $B$

The lamp-post subtends an angle of $3 0^{\circ}$ at $B$ .
This means $∠ P B D = 3 0^{\circ}$ .

Analyzing Right-Angled $△ P D B$

In $△ P D B$ , $tan (3 0^{\circ}) = \frac{Perpendicular}{Base}$ .
$tan (3 0^{\circ}) = \frac{D P}{B D}$ .

Expressing $B D$ in terms of $h$

$\frac{1}{3} = \frac{h}{B D}$
$⟹ B D = h 3$

Identifying $B D$ as the Median

In $△ A B C$ , $B D$ connects vertex $B$ to the midpoint of $A C$ .
Thus, $B D$ is a median.
We use Apollonius Theorem.

Applying Apollonius Theorem

$A B^{2} + B C^{2} = 2 (B D^{2} + A D^{2})$

Substituting Known Values

Substitute $A B = 7$ , $B C = 5$ , $A D = 3$ :
$7^{2} + 5^{2} = 2 (B D^{2} + 3^{2})$

Evaluating the Squares

$49 + 25 = 2 (B D^{2} + 9)$

Simplifying the Equation

$74 = 2 (B D^{2} + 9)$

Solving for $B D^{2}$

Divide by $2$ :
$37 = B D^{2} + 9$
Subtract $9$ :
$B D^{2} = 28$

Connecting $B D^{2}$ back to $h$

Recall $B D = h 3$ .
Squaring both sides:
$B D^{2} = 3 h^{2}$

Calculating the Height $h$

Substitute $B D^{2} = 28$ :
$3 h^{2} = 28$
$⟹ h^{2} = \frac{28}{3}$
$⟹ h = \frac{28}{3}$

Simplifying the Final Answer

$h = \frac{4 \times 7}{3} = \frac{2 7}{3}$
Rationalize the denominator:
$h = \frac{2 21}{3}$

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The Sigma Insight: Heights and Distances

Solution Diagram

Analyzing the Setup

We are given a triangular plot

A B C

with side lengths

A B = 7

B C = 5

, and

C A = 6

. A vertical lamp-post of height

h

stands at point

D

, which is the midpoint of side

A C

The lamp-post, denoted as

D P

, is perpendicular to the ground. We are given that the lamp-post subtends an angle of

3 0^{\circ}

at vertex

B

The 3D Perspective

Consider the right-angled triangle

△ P D B

, where

D P = h

is the height of the lamp-post and

B D

is the base on the ground. Using the definition of the tangent function:

tan (3 0^{\circ}) = \frac{D P}{B D}

Since

tan (3 0^{\circ}) = \frac{1}{3}

, we substitute the known values:

\frac{1}{3} = \frac{h}{B D} \Rightarrow B D = h 3

This relationship establishes the connection between the height of the lamp-post and the length of the median

B D

The Geometry of the Median

To find the length of

B D

, we focus on the geometry of

△ A B C

. Since

D

is the midpoint of

A C

B D

is the median to side

A C

We apply Apollonius Theorem, which relates the lengths of the sides of a triangle to the length of its median:

A B^{2} + B C^{2} = 2 (B D^{2} + A D^{2})

Given

A B = 7

B C = 5

, and

A C = 6

, the midpoint

D

implies

A D = \frac{A C}{2} = 3

Final Calculation

Substituting the known values into the Apollonius equation:

7^{2} + 5^{2} = 2 (B D^{2} + 3^{2})

49 + 25 = 2 (B D^{2} + 9)

74 = 2 (B D^{2} + 9)

Dividing by

2

and solving for

B D^{2}

37 = B D^{2} + 9 \Rightarrow B D^{2} = 28

Recall our earlier relationship

B D = h 3

, which implies

B D^{2} = 3 h^{2}

. Equating the two expressions for

B D^{2}

3 h^{2} = 28 \Rightarrow h^{2} = \frac{28}{3}

Taking the square root and rationalizing the denominator:

h = \frac{28}{3} = \frac{2 7}{3} \cdot \frac{3}{3} = \frac{2 21}{3}

The height of the lamp-post is $\frac{2 21}{3}$ meters.

Common Traps & FAQs

Why can't we just use the Sine Rule to find

B D

You absolutely could! But it would involve finding

cos (∠ C)

∠ A

first using the Cosine Rule, then finding the angle

∠ B

, and then applying the Sine Rule. It is a much longer, more winding path. In the JEE, time is currency. The Apollonius Theorem is the 'express lane' for medians. Always look for the most direct geometric property.

How do I know

D

is the midpoint?

The problem explicitly states

D

is the midpoint of

A C

. Always read the problem statement carefully—sometimes the most crucial piece of information is hidden in a simple sentence. If

D

were not the midpoint,

B D

would just be a cevian, and we would need Stewart's Theorem instead of Apollonius.

Is the lamp-post height

h

related to the angle

3 0^{\circ}

directly?

Yes, exactly. Because the lamp-post is vertical, it creates a right-angled triangle with the ground. The angle of elevation from the base of the triangle (

B

) to the top of the post (

P

) is

3 0^{\circ}

. Therefore,

tan (3 0^{\circ}) = \frac{height}{base}

, which gives us the vital link

B D = h 3

Animated Solution for Mathematics - Trigonometry: Consider a triangular plot ABC with sides AB=7m, BC=5m and CA=6m. A vertical lamp-post at the mid point D of AC subtends an angle $3 0^{\circ}$ at B. The height (in m) of the lamp-post is:

Select Answer:

Visualized Solution

Visualizing the Triangular Plot $A B C$

Locating the Lamp-post at Midpoint $D$

The Angle Subtended at $B$

Analyzing Right-Angled $△ P D B$

Expressing $B D$ in terms of $h$

Identifying $B D$ as the Median

Applying Apollonius Theorem

Substituting Known Values

Evaluating the Squares

Simplifying the Equation

Solving for $B D^{2}$

Connecting $B D^{2}$ back to $h$

Calculating the Height $h$

Simplifying the Final Answer

The Sigma Insight: Heights and Distances

Analyzing the Setup

The 3D Perspective

The Geometry of the Median

Final Calculation

Similar Questions

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Select Answer:

Visualized Solution

Visualizing the Triangular Plot .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABC

Locating the Lamp-post at Midpoint .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D

The Angle Subtended at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Analyzing Right-Angled .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } △PDB

Expressing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } BD in terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } h

Identifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } BD as the Median

Applying Apollonius Theorem

Substituting Known Values

Evaluating the Squares

Simplifying the Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } BD2

Connecting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } BD2 back to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } h

Calculating the Height .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } h

Simplifying the Final Answer

The Sigma Insight: Heights and Distances

Analyzing the Setup

The 3D Perspective

The Geometry of the Median

Final Calculation

Similar Questions

Visualizing the Triangular Plot $A B C$

Locating the Lamp-post at Midpoint $D$

The Angle Subtended at $B$

Analyzing Right-Angled $△ P D B$

Expressing $B D$ in terms of $h$

Identifying $B D$ as the Median

Solving for $B D^{2}$

Connecting $B D^{2}$ back to $h$

Calculating the Height $h$