Practice Questions

Define the term 'semi-perimeter' of a triangle.

Explain what each variable represents in the formula: Area = $\sqrt{s(s-a)(s-b)(s-c)}$

The perimeter of an equilateral triangle is 60 m. Calculate its area using Heron's formula.

The perimeter of a triangular field is 150 m. What is its semi-perimeter?

Calculate the area of a triangular field whose sides are 50 m, 78 m, and 112 m.

Justify when the formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ is more efficient to use for finding the area of a triangle compared to Heron's formula.

Calculate the semi-perimeter of a triangle whose sides measure 20 cm, 21 cm, and 29 cm.

State Heron's formula for finding the area of a triangle with sides of length $a, b$, and $c$.

For a triangle with sides $a, b, c$, write the expression used to calculate its semi-perimeter.

In Heron's formula, what does the variable 's' represent?

Name the type of triangle for which Heron's formula is most useful, especially when the height is not known.

A triangle has sides of length 13 cm, 14 cm, and 15 cm. Calculate the value of its semi-perimeter, $s$.

For a triangle with sides $a = 20$ cm, $b = 21$ cm, and $c = 29$ cm, find the values of $(s-a)$, $(s-b)$, and $(s-c)$.

List the steps required to find the area of a triangle using Heron's formula, given the lengths of its three sides.

Describe the complete process of finding the area of an isosceles triangle with equal sides of 10 cm and a base of 12 cm using Heron's formula. Write down each step and calculate the required values before the final area calculation.

A triangle has sides of 5 cm, 12 cm, and 13 cm. Analyze if it is a right-angled triangle and then calculate its area without using Heron's formula.

The perimeter of a triangle is 42 cm. If two of its sides are 13 cm and 14 cm, calculate its area.

A triangular park has sides 90 m, 70 m, and 40 m. Calculate the area available for planting saplings.

An isosceles triangle has a perimeter of 64 cm. If the length of its unequal side (base) is 14 cm, calculate the area of the triangle.

A triangular advertisement board has dimensions 13 m, 14 m, and 15 m. Calculate the cost of painting it at a rate of ₹12.50 per m².

The sides of a triangular field are in the ratio 12:17:25 and its perimeter is 1080 m. A farmer wants to plough the field at a rate of ₹50 per 100 m². Calculate the total cost of ploughing the field.

A student claims that Heron's formula is unnecessarily complicated and should not be used for right-angled triangles. Critique this statement.

Evaluate whether a triangle can be constructed with sides 4 cm, 6 cm, and 11 cm. Justify your answer in the context of applying Heron's formula.

Formulate a general expression for the area of an isosceles triangle with two equal sides '$a$' and a base of length '$b$' using Heron's formula, without performing the final simplification.

Derive the standard formula for the area of an equilateral triangle, $\text{Area} = \frac{\sqrt{3}}{4}a^2$, by starting with Heron's formula.

Evaluate the percentage increase in the area of a triangle if each of its sides is doubled. Justify your answer algebraically using Heron's formula.

A quadrilateral park ABCD has side lengths AB = 9 m, BC = 40 m, CD = 28 m, and DA = 15 m. The angle at B, $\angle ABC$, is a right angle ($90^\circ$). Formulate a plan to calculate the area of the park. Justify your steps and execute the plan to find the total area.

An isosceles triangular park has a perimeter of 80 m. The length of its base is 30 m. The park management wishes to find the shortest distance from the vertex between the equal sides to the base. Justify how Heron's formula can be used to find this distance (the altitude) and then calculate it.

For a triangle with sides a, b, and c, the semi-perimeter is 20 cm. If side a = 15 cm and side b = 10 cm, calculate the value of (s-c).

A rhombus has a perimeter of 52 m and one of its diagonals is 10 m. Justify how you can use Heron's formula to find the area of the rhombus and then calculate it.

An equilateral triangle has a perimeter of 120 m. Explain the steps to find its area using Heron's formula, and calculate the value of $s$ and the terms $(s-a)$, $(s-b)$, and $(s-c)$.

Apply Heron's formula to find the area of an equilateral triangle with each side measuring '2a' units.

The semi-perimeter of a triangle is 30 cm. If its sides are in the ratio 3:4:5, calculate the length of the longest side.

An old triangular manuscript states that its sides were in the ratio 5:12:13 and its area was 480 square units. A historian proposes that the actual side lengths were 20, 48, and 52 units. Evaluate the historian's proposal by first deriving the side lengths from the given information and then justifying your conclusion.

A park is in the shape of a quadrilateral ABCD, where $\angle C = 90^\circ$, AB = 9 m, BC = 12 m, CD = 5 m, and AD = 8 m. Calculate the total area the park occupies.

A rhombus-shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, analyze the grazing area to find how much area of the grass field each cow will get.

The semi-perimeter of an equilateral triangle is 24 cm. What is the length of each side of the triangle?

Design a non-right-angled triangle where the side lengths are consecutive integers and the area is also an integer. Justify that your designed triangle meets these criteria by calculating its area.

A triangle has side lengths of 8 m, 15 m, and 17 m. Calculate its area using Heron's formula. Show all the steps.

Propose a conceptual reason for the use of the semi-perimeter, $s$, in Heron's formula rather than the full perimeter, $P$.

Describe a situation where Heron's formula is more advantageous to use than the standard formula Area = $\frac{1}{2} \times \text{base} \times \text{height}$.

A triangular plot of land has sides 26 m, 28 m, and 30 m. The owner wants to divide it into two smaller triangular plots of equal area for his two children. He plans to do this by joining the midpoint of the longest side to the opposite vertex. Justify that this method is valid and then calculate the area of each smaller plot.

A field is in the shape of a trapezium ABCD, where AB is parallel to DC. The sides are AB = 25 m, DC = 10 m, BC = 14 m, and AD = 13 m. Formulate a strategy to find the area of this field by dividing it into a triangle and a parallelogram. Justify your method and calculate the total area.

Create a word problem involving a composite figure made of two triangles sharing a common side. The problem must require the use of Heron's formula for both triangles and ask for the total cost of fertilizing the area. Provide the complete solution for the problem you create.

Calculate the area of a quadrilateral ABCD in which AB = 9 cm, BC = 40 cm, CD = 28 cm, DA = 15 cm and the diagonal AC = 41 cm.