Finding the Area of Triangle PQR

Oct 13, 2025 by miodeya.com 116 views

Introduction

In geometry, finding the area of a figure is a crucial concept that helps us understand the size and shape of various shapes. In this problem, we are given a figure PQRS with specific side lengths and are asked to find its area. To solve this problem, we need to use the concept of area and apply it to the given figure.

Understanding the Figure

The figure PQRS is a quadrilateral with side lengths given as follows:

PQ = 14 cm
QR = 18 cm
RS = 12 cm
SP = 12 cm

Breaking Down the Figure

To find the area of the figure PQRS, we can break it down into smaller shapes. One way to do this is to divide the figure into two triangles: triangle PQR and triangle PSR.

Finding the Area of Triangle PQR

To find the area of triangle PQR, we need to use the formula for the area of a triangle, which is given by:

Area = (base × height) / 2

In this case, the base of triangle PQR is PQ = 14 cm, and the height can be found using the Pythagorean theorem. However, since we are not given the height directly, we can use the fact that triangle PQR is a right-angled triangle (since PQ = 14 cm and QR = 18 cm form a right angle).

Using the Pythagorean theorem, we can find the height of triangle PQR:

Height = √(QR² - PQ²) = √(18² - 14²) = √(324 - 196) = √128 = 11.3137 cm (approximately)

Now that we have the base and height of triangle PQR, we can find its area:

Area of triangle PQR = (base × height) / 2 = (14 × 11.3137) / 2 = 100.25 cm² (approximately)

Finding the Area of Triangle PSR

To find the area of triangle PSR, we need to use the formula for the area of a triangle, which is given by:

Area = (base × height) / 2

In this case, the base of triangle PSR is RS = 12 cm, and the height can be found using the Pythagorean theorem. However, since we are not given the height directly, we can use the fact that triangle PSR is a right-angled triangle (since RS = 12 cm and SP = 12 cm form a right angle).

Using the Pythagorean theorem, we can find the height of triangle PSR:

Height = √(SP² - RS²) = √(12² - 12²) = √(0) = 0 cm

However, this is not possible since the height of a triangle cannot be zero. This means that triangle PSR is actually a degenerate triangle, and its area is zero.

Finding the Area of the Figure PQRS

Now that we have found the areas of triangles PQR and PSR, we can find the area of the figure PQRS by adding the areas of the two triangles:

Area of figure PQRS = Area of triangle PQR + Area of triangle PSR = 100.25 cm² + 0 cm² = 100.25 cm²

Therefore, the area of the figure PQRS is approximately 100.25 cm².

Conclusion

In this problem, we were given a figure PQRS with specific side lengths and were asked to find its area. To solve this problem, we broke down the figure into smaller shapes, found the areas of the individual triangles, and added them together to find the area of the figure PQRS. This problem required us to use the concept of area and apply it to a given figure, which is an important skill in geometry.

Key Takeaways

To find the area of a figure, we can break it down into smaller shapes and find the areas of the individual shapes.
The area of a triangle can be found using the formula: Area = (base × height) / 2.
The Pythagorean theorem can be used to find the height of a triangle when the base and hypotenuse are given.
A degenerate triangle has an area of zero.

Practice Problems

Find the area of a triangle with a base of 10 cm and a height of 8 cm.
Find the area of a figure that consists of two triangles with areas of 50 cm² and 75 cm².
Find the area of a rectangle with a length of 15 cm and a width of 10 cm.

Solutions to Practice Problems

Area of triangle = (base × height) / 2 = (10 × 8) / 2 = 40 cm²
Area of figure = Area of triangle 1 + Area of triangle 2 = 50 cm² + 75 cm² = 125 cm²
Area of rectangle = length × width = 15 cm × 10 cm = 150 cm²

Introduction

In our previous article, we discussed how to find the area of the figure PQRS by breaking it down into smaller shapes and finding the areas of the individual triangles. However, we received many questions from readers who wanted to know more about the problem and how to approach it. In this article, we will answer some of the most frequently asked questions about finding the area of the figure PQRS.

Q: What is the formula for finding the area of a triangle?

A: The formula for finding the area of a triangle is:

Area = (base × height) / 2

This formula can be used to find the area of any triangle, as long as you know the base and height of the triangle.

Q: How do I find the height of a triangle when the base and hypotenuse are given?

A: To find the height of a triangle when the base and hypotenuse are given, you can use the Pythagorean theorem. The Pythagorean theorem states that:

a² + b² = c²

where a and b are the legs of the right triangle, and c is the hypotenuse. In this case, the base of the triangle is one of the legs, and the height is the other leg. You can rearrange the formula to solve for the height:

height = √(hypotenuse² - base²)

Q: What is a degenerate triangle?

A: A degenerate triangle is a triangle that has zero area. This can happen when the base and height of the triangle are both zero, or when the triangle is a line segment. In the case of the figure PQRS, triangle PSR is a degenerate triangle because it has a base and height of zero.

Q: How do I find the area of a figure that consists of multiple triangles?

A: To find the area of a figure that consists of multiple triangles, you can find the area of each individual triangle and add them together. This is known as the "sum of areas" method.

Q: What is the difference between the area of a triangle and the area of a rectangle?

A: The area of a triangle is found using the formula:

Area = (base × height) / 2

The area of a rectangle is found using the formula:

Area = length × width

The main difference between the two formulas is that the area of a triangle is divided by 2, while the area of a rectangle is not.

Q: Can I use the area of a triangle to find the area of a rectangle?

A: Yes, you can use the area of a triangle to find the area of a rectangle. If you know the area of a triangle and the length of one of its sides, you can use the formula:

Area of rectangle = (2 × area of triangle) / length

to find the area of the rectangle.

Q: What are some common mistakes to avoid when finding the area of a triangle?

A: Some common mistakes to avoid when finding the area of a triangle include:

Forgetting to divide the area by 2
Using the wrong formula for the area of a triangle
Not using the Pythagorean theorem to find the height of a triangle
Not checking for degenerate triangles

Q: How can I practice finding the area of triangles and figures?

A: There are many ways to practice finding the area of triangles and figures, including:

Using online resources and worksheets
Creating your own practice problems
Working with a tutor or teacher
Joining a study group or math club

Conclusion

In this article, we answered some of the most frequently asked questions about finding the area of the figure PQRS. We covered topics such as the formula for finding the area of a triangle, how to find the height of a triangle when the base and hypotenuse are given, and how to find the area of a figure that consists of multiple triangles. We also discussed common mistakes to avoid and how to practice finding the area of triangles and figures.