Question

A man of height 1.8m is standing near a pyramid. If the shadow of the man is of length 2.7m and the shadow of the pyramid is 210m long at that instant, find the height of the pyramid.

Answer 1

Hint: We draw a diagram for the given situation and use the concept of similar triangles. Prove the two triangles similar using AAA property and equate ratio of sides of two triangles.
* If two triangles are similar to each other, then the ratio of corresponding sides of triangles is the same for three corresponding sides.
* Two triangles are said to be similar if the triangles have congruent corresponding angles.

Step-By-Step answer:
We are given the height of man is 1.8m and length of man’s shadow is 2.7m. Also, we are given the length of the shadow of the pyramid is 210m and we have to calculate the height of the pyramid. We draw a diagram depicting the situation.

Here AB is the height of the pyramid and DE is the height of the man.
\[AB = x,AC = 210,ED = 1.8,EC = 2.7\] … (1)
We have both man and pyramid perpendicular to the ground, so the angles made by man and the pyramid with the ground will be right angles.
In \[\vartriangle ABC,\vartriangle EDC\]
\[\angle BAC = \angle DEC\] (As they both are corresponding angles and are equal i.e. \[{90^ \circ }\])
\[\angle ABC = \angle EDC\](As they both are corresponding angles formed by parallel lines AB and DE cut by transversal AC)
\[\angle C = \angle C\](Common angle)
By Angle Angle Angle (AAA) property we can say triangles\[\vartriangle ABC,\vartriangle EDC\] are similar to each other.
Now using the property of similar triangles, we can write the ratio of corresponding sides of triangles equal.
\[ \Rightarrow \dfrac{{AB}}{{ED}} = \dfrac{{AC}}{{EC}}\]
Substitute the values in the fraction from equation (1)
\[ \Rightarrow \dfrac{x}{{1.8}} = \dfrac{{210}}{{2.7}}\]
Multiply both sides by 1.8
\[ \Rightarrow \dfrac{x}{{1.8}} \times 1.8 = \dfrac{{210}}{{2.7}} \times 1.8\]
Cancel the same factors from numerator and denominator on both sides of the equation.
\[ \Rightarrow x = \dfrac{{210}}{9} \times 6\]
\[ \Rightarrow x = 140\]
\[ \Rightarrow AB = 140\]

\[\therefore \]The height of the pyramid is 140m.

Note: Alternate method:
Since we know both triangles are right triangles, we can apply the formula of trigonometry i.e. tangent of an angle.
We know tangent of an angle is given by dividing perpendicular by base.
In triangle ABC,
\[ \Rightarrow \tan C = \dfrac{{AB}}{{AC}}\]
Similarly in triangle DEC,
\[ \Rightarrow \tan C = \dfrac{{DE}}{{EC}}\]
Since tangent of an angle has finite value, we can equate values from both triangles.
\[ \Rightarrow \dfrac{{AB}}{{AC}} = \dfrac{{DE}}{{EC}}\]
Substitute values of \[AB = x,AC = 210,ED = 1.8,EC = 2.7\]
\[ \Rightarrow \dfrac{x}{{1.8}} = \dfrac{{210}}{{2.7}}\]
Multiply both sides by 1.8
\[ \Rightarrow \dfrac{x}{{1.8}} \times 1.8 = \dfrac{{210}}{{2.7}} \times 1.8\]
Cancel the same factors from the numerator and denominator on both sides of the equation.
\[ \Rightarrow x = \dfrac{{210}}{9} \times 6\]
\[ \Rightarrow x = 140\]
\[ \Rightarrow AB = 140\]
\[\therefore \]The height of the pyramid is 140m.